Polylogarithm

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https://en.wikipedia.org/wiki/Polylogarithm

In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Li_s(z) of order s and argument z. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams.

The polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special cases of the Lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions, nor with the offset logarithmic integral Li(z), which has the same notation without the subscript.

• Different polylogarithm functions in the complex plane
• 

  Li _-3(z)

• 

  Li _-2(z)

• 

  Li _-1(z)

• 

  Li₀(z)

• 

  Li₁(z)

• 

  Li₂(z)

• 

  Li₃(z)

The polylogarithm function is defined by a power series in z, which is also a Dirichlet series in s:

${\displaystyle {\mathrm{Li}}_s(z)=\sum _{k=1}^{\mathrm{\infty }}\frac{z^k}{k^s}=z+\frac{z^2}{2^s}+\frac{z^3}{3^s}+\cdots }$

This definition is valid for arbitrary complex order s and for all complex arguments z with |z| < 1; it can be extended to |z| ≥ 1 by the process of analytic continuation. (Here the denominator k^s is understood as exp(s ln k)). The special case s = 1 involves the ordinary natural logarithm, Li₁(z) = −ln(1−z), while the special cases s = 2 and s = 3 are called the dilogarithm (also referred to as Spence's function) and trilogarithm respectively. The name of the function comes from the fact that it may also be defined as the repeated integral of itself:

${\displaystyle {\mathrm{Li}}_{s+1}(z)={\int }_0^z\frac{{\mathrm{Li}}_s(t)}{t}dt}$

thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orders s, the polylogarithm is a rational function.

Properties

In the case where the order ${\displaystyle s}$ is an integer, it will be represented by ${\displaystyle s=n}$ (or ${\displaystyle s=-n}$ when negative). It is often convenient to define ${\displaystyle \mu =\ln (z)}$ where ${\displaystyle \ln (z)}$ is the principal branch of the complex logarithm ${\displaystyle \mathrm{Ln}(z)}$ so that ${\displaystyle -\pi <\mathrm{Im}(\mu )\le \pi .}$ Also, all exponentiation will be assumed to be single-valued: ${\displaystyle z^s=\exp (s\ln (z)).}$

Depending on the order ${\displaystyle s}$, the polylogarithm may be multi-valued. The principal branch of ${\displaystyle {\mathrm{Li}}_s(z)}$ is taken to be given for ${\displaystyle |z|<1}$ by the above series definition and taken to be continuous except on the positive real axis, where a cut is made from ${\displaystyle z=1}$ to ${\displaystyle \mathrm{\infty }}$ such that the axis is placed on the lower half plane of ${\displaystyle z}$. In terms of ${\displaystyle \mu }$, this amounts to ${\displaystyle -\pi <\arg (-\mu )\le \pi }$. The discontinuity of the polylogarithm in dependence on ${\displaystyle \mu }$ can sometimes be confusing.

For real argument ${\displaystyle z}$, the polylogarithm of real order ${\displaystyle s}$ is real if ${\displaystyle z<1}$, and its imaginary part for ${\displaystyle z\ge 1}$ is (Wood 1992, § 3):

$${\displaystyle \mathrm{Im}\left({\mathrm{Li}}_s(z)\right)=-\frac{\pi {\mu }^{s-1}}{\mathrm{\Gamma }(s)}.}$$

Going across the cut, if ε is an infinitesimally small positive real number, then:

$${\displaystyle \mathrm{Im}\left({\mathrm{Li}}_s(z+iϵ)\right)=\frac{\pi {\mu }^{s-1}}{\mathrm{\Gamma }(s)}.}$$

Both can be concluded from the series expansion (see below) of Li_s(e^µ) about µ = 0.

The derivatives of the polylogarithm follow from the defining power series:

$${\displaystyle z\frac{\mathrm{\partial }{\mathrm{Li}}_s(z)}{\mathrm{\partial }z}={\mathrm{Li}}_{s-1}(z)}$$

$${\displaystyle \frac{\mathrm{\partial }{\mathrm{Li}}_s(e^{\mu })}{\mathrm{\partial }\mu }={\mathrm{Li}}_{s-1}(e^{\mu }).}$$

The square relationship is seen from the series definition, and is related to the duplication formula (see also Clunie (1954), Schrödinger (1952)):

$${\displaystyle {\mathrm{Li}}_s(-z)+{\mathrm{Li}}_s(z)=2^{1-s}{\mathrm{Li}}_s(z^2).}$$

Kummer's function obeys a very similar duplication formula. This is a special case of the multiplication formula, for any positive integer p:

$${\displaystyle \sum _{m=0}^{p-1}{\mathrm{Li}}_s(z{e}^{2\pi im/p})=p^{1-s}{\mathrm{Li}}_s(z^p),}$$

which can be proved using the series definition of the polylogarithm and the orthogonality of the exponential terms (see e.g. discrete Fourier transform).

Another important property, the inversion formula, involves the Hurwitz zeta function or the Bernoulli polynomials and is found under relationship to other functions below.

Particular values

For particular cases, the polylogarithm may be expressed in terms of other functions (see below). Particular values for the polylogarithm may thus also be found as particular values of these other functions.

1. For integer values of the polylogarithm order, the following explicit expressions are obtained by repeated application of z·∂/∂z to Li₁(z):
   $${\displaystyle {\mathrm{Li}}_1(z)=-\ln (1-z)}$$

   
   $${\displaystyle {\mathrm{Li}}_0(z)=\frac{z}{1-z}}$$

   
   $${\displaystyle {\mathrm{Li}}_{-1}(z)=\frac{z}{(1-z{)}^2}}$$

   
   $${\displaystyle {\mathrm{Li}}_{-2}(z)=\frac{z(1+z)}{(1-z{)}^3}}$$

   
   $${\displaystyle {\mathrm{Li}}_{-3}(z)=\frac{z(1+4z+z^2)}{(1-z{)}^4}}$$

   
   $${\displaystyle {\mathrm{Li}}_{-4}(z)=\frac{z(1+z)(1+10z+z^2)}{(1-z{)}^5}.}$$

   
   Accordingly the polylogarithm reduces to a ratio of polynomials in z, and is therefore a rational function of z, for all nonpositive integer orders. The general case may be expressed as a finite sum:
   $${\displaystyle {\mathrm{Li}}_{-n}(z)={\left(z\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)}^n\frac{z}{1-z}=\sum _{k=0}^{n}k!S(n+1,k+1){\left(\frac{z}{1-z}\right)}^{k+1}(n=0,1,2,\dots ),}$$

   
   where S(n,k) are the Stirling numbers of the second kind. Equivalent formulae applicable to negative integer orders are (Wood 1992, § 6):
   $${\displaystyle {\mathrm{Li}}_{-n}(z)=(-1{)}^{n+1}\sum _{k=0}^{n}k!S(n+1,k+1){\left(\frac{-1}{1-z}\right)}^{k+1}(n=1,2,3,\dots ),}$$

   
   and:
   $${\displaystyle {\mathrm{Li}}_{-n}(z)=\frac{1}{(1-z{)}^{n+1}}\sum _{k=0}^{n-1}⟨\genfrac{}{}{0ex}{}{n}{k}⟩z^{n-k}(n=1,2,3,\dots ),}$$

   
   where ${\displaystyle ⟨\genfrac{}{}{0ex}{}{n}{k}⟩}$ are the Eulerian numbers. All roots of Li_−n(z) are distinct and real; they include z = 0, while the remainder is negative and centered about z = −1 on a logarithmic scale. As n becomes large, the numerical evaluation of these rational expressions increasingly suffers from cancellation (Wood 1992, § 6); full accuracy can be obtained, however, by computing Li_−n(z) via the general relation with the Hurwitz zeta function (see below).
2. Some particular expressions for half-integer values of the argument z are:
   $${\displaystyle {\mathrm{Li}}_1(\frac{1}{2})=\ln 2}$$

   
   $${\displaystyle {\mathrm{Li}}_2(\frac{1}{2})=\frac{1}{12}{\pi }^2-\frac{1}{2}(\ln 2{)}^2}$$

   
   $${\displaystyle {\mathrm{Li}}_3(\frac{1}{2})=\frac{1}{6}(\ln 2{)}^3-\frac{1}{12}{\pi }^2\ln 2+\frac{7}{8}\zeta (3),}$$

   
   where ζ is the Riemann zeta function. No formulae of this type are known for higher integer orders (Lewin 1991, p. 2), but one has for instance (Borwein, Borwein & Girgensohn 1995):
   $${\displaystyle {\mathrm{Li}}_4(\frac{1}{2})=\frac{1}{360}{\pi }^4-\frac{1}{24}(\ln 2{)}^4+\frac{1}{24}{\pi }^2(\ln 2{)}^2-\frac{1}{2}\zeta (\overline{3},\overline{1}),}$$

   
   which involves the alternating double sum
   $${\displaystyle \zeta (\overline{3},\overline{1})=\sum _{m>n>0}(-1{)}^{m+n}{m}^{-3}{n}^{-1}.}$$

   
   In general one has for integer orders n ≥ 2 (Broadhurst 1996, p. 9):
   $${\displaystyle {\mathrm{Li}}_n(\frac{1}{2})=-\zeta (\overline{1},\overline{1},{\left\{1\right\}}^{n-2}),}$$

   
   where ζ(s₁, …, s_k) is the multiple zeta function; for example:
   $${\displaystyle {\mathrm{Li}}_5(\frac{1}{2})=-\zeta (\overline{1},\overline{1},1,1,1).}$$

   
3. As a straightforward consequence of the series definition, values of the polylogarithm at the pth complex roots of unity are given by the Fourier sum:
   $${\displaystyle {\mathrm{Li}}_s(e^{2\pi im/p})=p^{-s}\sum _{k=1}^p{e}^{2\pi imk/p}\zeta (s,\frac{k}{p})(m=1,2,\dots ,p-1),}$$

   
   where ζ is the Hurwitz zeta function. For Re(s) > 1, where Li_s(1) is finite, the relation also holds with m = 0 or m = p. While this formula is not as simple as that implied by the more general relation with the Hurwitz zeta function listed under relationship to other functions below, it has the advantage of applying to non-negative integer values of s as well. As usual, the relation may be inverted to express ζ(s, ^m⁄_p) for any m = 1, …, p as a Fourier sum of Li_s(exp(2πi ^k⁄_p)) over k = 1, …, p.

Relationship to other functions

• For z = 1, the polylogarithm reduces to the Riemann zeta function
  $${\displaystyle {\mathrm{Li}}_s(1)=\zeta (s)(\mathrm{Re}(s)>1).}$$

  
• The polylogarithm is related to Dirichlet eta function and the Dirichlet beta function:
  $${\displaystyle {\mathrm{Li}}_s(-1)=-\eta (s),}$$

  
  where η(s) is the Dirichlet eta function. For pure imaginary arguments, we have:
  $${\displaystyle {\mathrm{Li}}_s(\pm i)=-2^{-s}\eta (s)\pm i\beta (s),}$$

  
  where β(s) is the Dirichlet beta function.
• The polylogarithm is related to the complete Fermi–Dirac integral as:
  $${\displaystyle F_s(\mu )=-{\mathrm{Li}}_{s+1}(-e^{\mu }).}$$

  
• The polylogarithm is a special case of the incomplete polylogarithm function
  $${\displaystyle {\mathrm{Li}}_s(z)={\mathrm{Li}}_s(0,z).}$$

  
• The polylogarithm is a special case of the Lerch transcendent (Erdélyi et al. 1981, § 1.11-14)
  $${\displaystyle {\mathrm{Li}}_s(z)=z\mathrm{\Phi }(z,s,1).}$$

  
• The polylogarithm is related to the Hurwitz zeta function by:

$${\displaystyle {\mathrm{Li}}_s(z)=\frac{\mathrm{\Gamma }(1-s)}{(2\pi {)}^{1-s}}\left[i^{1-s}\zeta \left(1-s,\frac{1}{2}+\frac{\ln (-z)}{2\pi i}\right)+i^{s-1}\zeta \left(1-s,\frac{1}{2}-\frac{\ln (-z)}{2\pi i}\right)\right],}$$

which relation, however, is invalidated at positive integer s by poles of the gamma function Γ(1 − s), and at s = 0 by a pole of both zeta functions; a derivation of this formula is given under series representations below. With a little help from a functional equation for the Hurwitz zeta function, the polylogarithm is consequently also related to that function via (Jonquière 1889):

$${\displaystyle i^{-s}{\mathrm{Li}}_s(e^{2\pi ix})+i^s{\mathrm{Li}}_s(e^{-2\pi ix})=\frac{(2\pi {)}^s}{\mathrm{\Gamma }(s)}\zeta (1-s,x),}$$

which relation holds for 0 ≤ Re(x) < 1 if Im(x) ≥ 0, and for 0 < Re(x) ≤ 1 if Im(x) < 0. Equivalently, for all complex s and for complex z ∉ ]0;1], the inversion formula reads

$${\displaystyle {\mathrm{Li}}_s(z)+(-1{)}^s{\mathrm{Li}}_s(1/z)=\frac{(2\pi i{)}^s}{\mathrm{\Gamma }(s)}\zeta \left(1-s,\frac{1}{2}+\frac{\ln (-z)}{2\pi i}\right),}$$

and for all complex s and for complex z ∉ ]1;∞[

$${\displaystyle {\mathrm{Li}}_s(z)+(-1{)}^s{\mathrm{Li}}_s(1/z)=\frac{(2\pi i{)}^s}{\mathrm{\Gamma }(s)}\zeta \left(1-s,\frac{1}{2}-\frac{\ln (-1/z)}{2\pi i}\right).}$$

For z ∉ ]0;∞[, one has ln(−z) = −ln(−¹⁄_z), and both expressions agree. These relations furnish the analytic continuation of the polylogarithm beyond the circle of convergence |z| = 1 of the defining power series. (The corresponding equation of Jonquière (1889, eq. 5) and Erdélyi et al. (1981, § 1.11-16) is not correct if one assumes that the principal branches of the polylogarithm and the logarithm are used simultaneously.) See the next item for a simplified formula when s is an integer.

• For positive integer polylogarithm orders s, the Hurwitz zeta function ζ(1−s, x) reduces to Bernoulli polynomials, ζ(1−n, x) = −B_n(x) / n, and Jonquière's inversion formula for n = 1, 2, 3, … becomes:

$${\displaystyle {\mathrm{Li}}_n(e^{2\pi ix})+(-1{)}^n{\mathrm{Li}}_n(e^{-2\pi ix})=-\frac{(2\pi i{)}^n}{n!}{B}_n(x),}$$

where again 0 ≤ Re(x) < 1 if Im(x) ≥ 0, and 0 < Re(x) ≤ 1 if Im(x) < 0. Upon restriction of the polylogarithm argument to the unit circle, Im(x) = 0, the left hand side of this formula simplifies to 2 Re(Li_n(e^2πix)) if n is even, and to 2i Im(Li_n(e^2πix)) if n is odd. For negative integer orders, on the other hand, the divergence of Γ(s) implies for all z that (Erdélyi et al. 1981, § 1.11-17):

$${\displaystyle {\mathrm{Li}}_{-n}(z)+(-1{)}^n{\mathrm{Li}}_{-n}(1/z)=0(n=1,2,3,\dots ).}$$

More generally, one has for n = 0, ±1, ±2, ±3, …:

$${\displaystyle {\mathrm{Li}}_n(z)+(-1{)}^n{\mathrm{Li}}_n(1/z)=-\frac{(2\pi i{)}^n}{n!}{B}_n\left(\frac{1}{2}+\frac{\ln (-z)}{2\pi i}\right)(z\notin ]0;1]),}$$

$${\displaystyle {\mathrm{Li}}_n(z)+(-1{)}^n{\mathrm{Li}}_n(1/z)=-\frac{(2\pi i{)}^n}{n!}{B}_n\left(\frac{1}{2}-\frac{\ln (-1/z)}{2\pi i}\right)(z\notin ]1;\mathrm{\infty }[),}$$

where both expressions agree for z ∉ ]0;∞[. (The corresponding equation of Jonquière (1889, eq. 1) and Erdélyi et al. (1981, § 1.11-18) is again not correct.)

• The polylogarithm with pure imaginary μ may be expressed in terms of the Clausen functions Ci_s(θ) and Si_s(θ), and vice versa (Lewin 1958, Ch. VII § 1.4; Abramowitz & Stegun 1972, § 27.8):

$${\displaystyle {\mathrm{Li}}_s(e^{\pm i\theta })=C{i}_s(\theta )\pm iS{i}_s(\theta ).}$$

• The inverse tangent integral Ti_s(z) (Lewin 1958, Ch. VII § 1.2) can be expressed in terms of polylogarithms:

$${\displaystyle {\mathrm{Ti}}_s(z)=\frac{1}{2i}\left[{\mathrm{Li}}_s(iz)-{\mathrm{Li}}_s(-iz)\right].}$$

The relation in particular implies:

$${\displaystyle {\mathrm{Ti}}_0(z)=\frac{z}{1+z^2},{\mathrm{Ti}}_1(z)=\arctan z,{\mathrm{Ti}}_2(z)={\int }_0^z\frac{\arctan t}{t}dt,\dots {\mathrm{Ti}}_{n+1}(z)={\int }_0^z\frac{{\mathrm{Ti}}_n(t)}{t}dt,}$$

which explains the function name.

• The Legendre chi function χ_s(z) (Lewin 1958, Ch. VII § 1.1; Boersma & Dempsey 1992) can be expressed in terms of polylogarithms:

$${\displaystyle {\chi }_s(z)=\frac{1}{2}\left[{\mathrm{Li}}_s(z)-{\mathrm{Li}}_s(-z)\right].}$$

• The polylogarithm of integer order can be expressed as a generalized hypergeometric function:

$${\displaystyle {\mathrm{Li}}_n(z)=z_{n+1}{F}_n(1,1,\dots ,1;2,2,\dots ,2;z)(n=0,1,2,\dots ),}$$

$${\displaystyle {\mathrm{Li}}_{-n}(z)=z_n{F}_{n-1}(2,2,\dots ,2;1,1,\dots ,1;z)(n=1,2,3,\dots ).}$$

• In terms of the incomplete zeta functions or "Debye functions" (Abramowitz & Stegun 1972, § 27.1):

$${\displaystyle Z_n(z)=\frac{1}{(n-1)!}{\int }_z^{\mathrm{\infty }}\frac{t^{n-1}}{e^t-1}dt(n=1,2,3,\dots ),}$$

the polylogarithm Li_n(z) for positive integer n may be expressed as the finite sum (Wood 1992, § 16):

$${\displaystyle {\mathrm{Li}}_n(e^{\mu })=\sum _{k=0}^{n-1}{Z}_{n-k}(-\mu )\frac{{\mu }^k}{k!}(n=1,2,3,\dots ).}$$

A remarkably similar expression relates the "Debye functions" Z_n(z) to the polylogarithm:

$${\displaystyle Z_n(z)=\sum _{k=0}^{n-1}{\mathrm{Li}}_{n-k}(e^{-z})\frac{z^k}{k!}(n=1,2,3,\dots ).}$$

• Using Lambert series, if ${\displaystyle J_s(n)}$ is Jordan's totient function, then

$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{z^n{J}_{-s}(n)}{1-z^n}={\mathrm{Li}}_s(z).}$$

Integral representations

Any of the following integral representations furnishes the analytic continuation of the polylogarithm beyond the circle of convergence |z| = 1 of the defining power series.

1. The polylogarithm can be expressed in terms of the integral of the Bose–Einstein distribution:
   $${\displaystyle {\mathrm{Li}}_s(z)=\frac{1}{\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}\frac{t^{s-1}}{e^t/z-1}dt.}$$

   
   This converges for Re(s) > 0 and all z except for z real and ≥ 1. The polylogarithm in this context is sometimes referred to as a Bose integral but more commonly as a Bose–Einstein integral. Similarly, the polylogarithm can be expressed in terms of the integral of the Fermi–Dirac distribution:
   $${\displaystyle -{\mathrm{Li}}_s(-z)=\frac{1}{\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}\frac{t^{s-1}}{e^t/z+1}dt.}$$

   
   This converges for Re(s) > 0 and all z except for z real and ≤ −1. The polylogarithm in this context is sometimes referred to as a Fermi integral or a Fermi–Dirac integral (GSL 2010). These representations are readily verified by Taylor expansion of the integrand with respect to z and termwise integration. The papers of Dingle contain detailed investigations of both types of integrals. The polylogarithm is also related to the integral of the Maxwell–Boltzmann distribution:
   $${\displaystyle \underset{z\to 0}{lim}\frac{{\mathrm{Li}}_s(z)}{z}=\frac{1}{\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}{t}^{s-1}{e}^{-t}dt=1.}$$

   
   This also gives the asymptotic behavior of polylogarithm at the vicinity of origin.
2. A complementary integral representation applies to Re(s) < 0 and to all z except to z real and ≥ 0:
   $${\displaystyle {\mathrm{Li}}_s(z)={\int }_0^{\mathrm{\infty }}\frac{t^{-s}\sin [s\pi /2-t\ln (-z)]}{\sinh (\pi t)}dt.}$$

   
   This integral follows from the general relation of the polylogarithm with the Hurwitz zeta function (see above) and a familiar integral representation of the latter.
3. The polylogarithm may be quite generally represented by a Hankel contour integral (Whittaker & Watson 1927, § 12.22, § 13.13), which extends the Bose–Einstein representation to negative orders s. As long as the t = μ pole of the integrand does not lie on the non-negative real axis, and s ≠ 1, 2, 3, …, we have:
   $${\displaystyle {\mathrm{Li}}_s(e^{\mu })=-\frac{\mathrm{\Gamma }(1-s)}{2\pi i}{\oint }_H\frac{(-t{)}^{s-1}}{e^{t-\mu }-1}dt}$$

   
   where H represents the Hankel contour. The integrand has a cut along the real axis from zero to infinity, with the axis belonging to the lower half plane of t. The integration starts at +∞ on the upper half plane (Im(t) > 0), circles the origin without enclosing any of the poles t = µ + 2kπi, and terminates at +∞ on the lower half plane (Im(t) < 0). For the case where µ is real and non-negative, we can simply subtract the contribution of the enclosed t = µ pole:
   $${\displaystyle {\mathrm{Li}}_s(e^{\mu })=-\frac{\mathrm{\Gamma }(1-s)}{2\pi i}{\oint }_H\frac{(-t{)}^{s-1}}{e^{t-\mu }-1}dt-2\pi iR}$$

   
   where R is the residue of the pole:
   $${\displaystyle R=\frac{i}{2\pi }\mathrm{\Gamma }(1-s)(-\mu {)}^{s-1}.}$$

   
4. When the Abel–Plana formula is applied to the defining series of the polylogarithm, a Hermite-type integral representation results that is valid for all complex z and for all complex s:
   $${\displaystyle {\mathrm{Li}}_s(z)=\frac{1}{2}z+\frac{\mathrm{\Gamma }(1-s,-\ln z)}{(-\ln z{)}^{1-s}}+2z{\int }_0^{\mathrm{\infty }}\frac{\sin (s\arctan t-t\ln z)}{(1+t^2{)}^{s/2}(e^{2\pi t}-1)}dt}$$

   
   where Γ is the upper incomplete gamma-function. All (but not part) of the ln(z) in this expression can be replaced by −ln(¹⁄_z). A related representation which also holds for all complex s,
   $${\displaystyle {\mathrm{Li}}_s(z)=\frac{1}{2}z+z{\int }_0^{\mathrm{\infty }}\frac{\sin [s\arctan t-t\ln (-z)]}{(1+t^2{)}^{s/2}\sinh (\pi t)}dt,}$$

   
   avoids the use of the incomplete gamma function, but this integral fails for z on the positive real axis if Re(s) ≤ 0. This expression is found by writing 2^s Li_s(−z) / (−z) = Φ(z², s, ¹⁄₂) − z Φ(z², s, 1), where Φ is the Lerch transcendent, and applying the Abel–Plana formula to the first Φ series and a complementary formula that involves 1 / (e^2πt + 1) in place of 1 / (e^2πt − 1) to the second Φ series.
5. As cited in, we can express an integral for the polylogarithm by integrating the ordinary geometric series termwise for ${\displaystyle s\in \mathbb{N}}$ as
   $${\displaystyle {\mathrm{Li}}_{s+1}(z)=\frac{z\cdot (-1{)}^s}{s!}{\int }_0^1\frac{{\log }^s(t)}{1-tz}dt.}$$

   

Series representations

1. As noted under integral representations above, the Bose–Einstein integral representation of the polylogarithm may be extended to negative orders s by means of Hankel contour integration:
   $${\displaystyle {\mathrm{Li}}_s(e^{\mu })=-\frac{\mathrm{\Gamma }(1-s)}{2\pi i}{\oint }_H\frac{(-t{)}^{s-1}}{e^{t-\mu }-1}dt,}$$

   
   where H is the Hankel contour, s ≠ 1, 2, 3, …, and the t = μ pole of the integrand does not lie on the non-negative real axis. The contour can be modified so that it encloses the poles of the integrand at t − µ = 2kπi, and the integral can be evaluated as the sum of the residues (Wood 1992, § 12, 13; Gradshteyn & Ryzhik 1980, § 9.553):
   $${\displaystyle {\mathrm{Li}}_s(e^{\mu })=\mathrm{\Gamma }(1-s)\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}(2k\pi i-\mu {)}^{s-1}.}$$

   
   This will hold for Re(s) < 0 and all μ except where e^μ = 1. For 0 < Im(µ) ≤ 2π the sum can be split as:
   $${\displaystyle {\mathrm{Li}}_s(e^{\mu })=\mathrm{\Gamma }(1-s)\left[(-2\pi i{)}^{s-1}\sum _{k=0}^{\mathrm{\infty }}{\left(k+\frac{\mu }{2\pi i}\right)}^{s-1}+(2\pi i{)}^{s-1}\sum _{k=0}^{\mathrm{\infty }}{\left(k+1-\frac{\mu }{2\pi i}\right)}^{s-1}\right],}$$

   
   where the two series can now be identified with the Hurwitz zeta function:
   $${\displaystyle {\mathrm{Li}}_s(e^{\mu })=\frac{\mathrm{\Gamma }(1-s)}{(2\pi {)}^{1-s}}\left[i^{1-s}\zeta \left(1-s,\frac{\mu }{2\pi i}\right)+i^{s-1}\zeta \left(1-s,1-\frac{\mu }{2\pi i}\right)\right](0<\mathrm{Im}(\mu )\le 2\pi ).}$$

   
   This relation, which has already been given under relationship to other functions above, holds for all complex s ≠ 0, 1, 2, 3, … and was first derived in (Jonquière 1889, eq. 6).
2. In order to represent the polylogarithm as a power series about µ = 0, we write the series derived from the Hankel contour integral as:
   $${\displaystyle {\mathrm{Li}}_s(e^{\mu })=\mathrm{\Gamma }(1-s)(-\mu {)}^{s-1}+\mathrm{\Gamma }(1-s)\sum _{h=1}^{\mathrm{\infty }}\left[(-2h\pi i-\mu {)}^{s-1}+(2h\pi i-\mu {)}^{s-1}\right].}$$

   
   When the binomial powers in the sum are expanded about µ = 0 and the order of summation is reversed, the sum over h can be expressed in closed form:
   $${\displaystyle {\mathrm{Li}}_s(e^{\mu })=\mathrm{\Gamma }(1-s)(-\mu {)}^{s-1}+\sum _{k=0}^{\mathrm{\infty }}\frac{\zeta (s-k)}{k!}{\mu }^k.}$$

   
   This result holds for |µ| < 2π and, thanks to the analytic continuation provided by the zeta functions, for all s ≠ 1, 2, 3, … . If the order is a positive integer, s = n, both the term with k = n − 1 and the gamma function become infinite, although their sum does not. One obtains (Wood 1992, § 9; Gradshteyn & Ryzhik 1980, § 9.554):
   $${\displaystyle \underset{s\to k+1}{lim}\left[\frac{\zeta (s-k)}{k!}{\mu }^k+\mathrm{\Gamma }(1-s)(-\mu {)}^{s-1}\right]=\frac{{\mu }^k}{k!}\left[\sum _{h=1}^k\frac{1}{h}-\ln (-\mu )\right],}$$

   
   where the sum over h vanishes if k = 0. So, for positive integer orders and for |μ| < 2π we have the series:
   $${\displaystyle {\mathrm{Li}}_n(e^{\mu })=\frac{{\mu }^{n-1}}{(n-1)!}\left[H_{n-1}-\ln (-\mu )\right]+\sum _{k=0,k\ne n-1}^{\mathrm{\infty }}\frac{\zeta (n-k)}{k!}{\mu }^k,}$$

   
   where H_n denotes the nth harmonic number:
   $${\displaystyle H_n=\sum _{h=1}^n\frac{1}{h},H_0=0.}$$

   
   The problem terms now contain −ln(−μ) which, when multiplied by μⁿ⁻¹, will tend to zero as μ → 0, except for n = 1. This reflects the fact that Li_s(z) exhibits a true logarithmic singularity at s = 1 and z = 1 since:
   $${\displaystyle \underset{\mu \to 0}{lim}\mathrm{\Gamma }(1-s)(-\mu {)}^{s-1}=0(\mathrm{Re}(s)>1).}$$

   
   For s close, but not equal, to a positive integer, the divergent terms in the expansion about µ = 0 can be expected to cause computational difficulties (Wood 1992, § 9). Erdélyi's corresponding expansion (Erdélyi et al. 1981, § 1.11-15) in powers of ln(z) is not correct if one assumes that the principal branches of the polylogarithm and the logarithm are used simultaneously, since ln(¹⁄_z) is not uniformly equal to −ln(z). For nonpositive integer values of s, the zeta function ζ(s − k) in the expansion about µ = 0 reduces to Bernoulli numbers: ζ(−n − k) = −B_1+n+k / (1 + n + k). Numerical evaluation of Li_−n(z) by this series does not suffer from the cancellation effects that the finite rational expressions given under particular values above exhibit for large n.
3. By use of the identity
   $${\displaystyle 1=\frac{1}{\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{s-1}dt(\mathrm{Re}(s)>0),}$$

   
   the Bose–Einstein integral representation of the polylogarithm (see above) may be cast in the form:
   $${\displaystyle {\mathrm{Li}}_s(z)=\frac{1}{2}z+\frac{z}{2\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{s-1}\coth \frac{t-\ln z}{2}dt(\mathrm{Re}(s)>0).}$$

   
   Replacing the hyperbolic cotangent with a bilateral series,
   $${\displaystyle \coth \frac{t-\ln z}{2}=2\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{2k\pi i+t-\ln z},}$$

   
   then reversing the order of integral and sum, and finally identifying the summands with an integral representation of the upper incomplete gamma function, one obtains:
   $${\displaystyle {\mathrm{Li}}_s(z)=\frac{1}{2}z+\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }(1-s,2k\pi i-\ln z)}{(2k\pi i-\ln z{)}^{1-s}}.}$$

   
   For both the bilateral series of this result and that for the hyperbolic cotangent, symmetric partial sums from −k_max to k_max converge unconditionally as k_max → ∞. Provided the summation is performed symmetrically, this series for Li_s(z) thus holds for all complex s as well as all complex z.
4. Introducing an explicit expression for the Stirling numbers of the second kind into the finite sum for the polylogarithm of nonpositive integer order (see above) one may write:
   $${\displaystyle {\mathrm{Li}}_{-n}(z)=\sum _{k=0}^n{\left(\frac{-z}{1-z}\right)}^{k+1}\sum _{j=0}^k(-1{)}^{j+1}(\genfrac{}{}{0ex}{}{k}{j})(j+1{)}^n(n=0,1,2,\dots ).}$$

   
   The infinite series obtained by simply extending the outer summation to ∞ (Guillera & Sondow 2008, Theorem 2.1):
   $${\displaystyle {\mathrm{Li}}_s(z)=\sum _{k=0}^{\mathrm{\infty }}{\left(\frac{-z}{1-z}\right)}^{k+1}\sum _{j=0}^k(-1{)}^{j+1}(\genfrac{}{}{0ex}{}{k}{j})(j+1{)}^{-s},}$$

   
   turns out to converge to the polylogarithm for all complex s and for complex z with Re(z) < ¹⁄₂, as can be verified for |^−z⁄_(1−z)| < ¹⁄₂ by reversing the order of summation and using:
   $${\displaystyle \sum _{k=j}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k}{j}){\left(\frac{-z}{1-z}\right)}^{k+1}={\left[{\left(\frac{-z}{1-z}\right)}^{-1}-1\right]}^{-j-1}=(-z{)}^{j+1}.}$$

   
   The inner coefficients of these series can be expressed by Stirling-number-related formulas involving the generalized harmonic numbers. For example, see generating function transformations to find proofs (references to proofs) of the following identities:
   $${\displaystyle \begin{array}{rl}{\mathrm{Li}}_2(z)& =\sum _{j\ge 1}\frac{(-1{)}^{j-1}}{2}\left(H_j^2+H_j^{(2)}\right)\frac{z^j}{(1-z{)}^{j+1}}\\ {\mathrm{Li}}_3(z)& =\sum _{j\ge 1}\frac{(-1{)}^{j-1}}{6}\left(H_j^3+3H_j{H}_j^{(2)}+2H_j^{(3)}\right)\frac{z^j}{(1-z{)}^{j+1}}.\end{array}}$$

   
   For the other arguments with Re(z) < ¹⁄₂ the result follows by analytic continuation. This procedure is equivalent to applying Euler's transformation to the series in z that defines the polylogarithm.

Asymptotic expansions

For |z| ≫ 1, the polylogarithm can be expanded into asymptotic series in terms of ln(−z):

$${\displaystyle {\mathrm{Li}}_s(z)=\frac{\pm i\pi }{\mathrm{\Gamma }(s)}[\ln (-z)\pm i\pi {]}^{s-1}-\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k(2\pi {)}^{2k}\frac{B_{2k}}{(2k)!}\frac{[\ln (-z)\pm i\pi {]}^{s-2k}}{\mathrm{\Gamma }(s+1-2k)},}$$

$${\displaystyle {\mathrm{Li}}_s(z)=\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k(1-2^{1-2k})(2\pi {)}^{2k}\frac{B_{2k}}{(2k)!}\frac{[\ln (-z){]}^{s-2k}}{\mathrm{\Gamma }(s+1-2k)},}$$

where B_2k are the Bernoulli numbers. Both versions hold for all s and for any arg(z). As usual, the summation should be terminated when the terms start growing in magnitude. For negative integer s, the expansions vanish entirely; for non-negative integer s, they break off after a finite number of terms. Wood (1992, § 11) describes a method for obtaining these series from the Bose–Einstein integral representation (his equation 11.2 for Li_s(e^µ) requires −2π < Im(µ) ≤ 0).

Limiting behavior

The following limits result from the various representations of the polylogarithm (Wood 1992, § 22):

$${\displaystyle \underset{|z|\to 0}{lim}{\mathrm{Li}}_s(z)=z}$$

$${\displaystyle \underset{|\mu |\to 0}{lim}{\mathrm{Li}}_s(e^{\mu })=\mathrm{\Gamma }(1-s)(-\mu {)}^{s-1}(\mathrm{Re}(s)<1)}$$

$${\displaystyle \underset{\mathrm{Re}(\mu )\to \mathrm{\infty }}{lim}{\mathrm{Li}}_s(\pm e^{\mu })=-\frac{{\mu }^s}{\mathrm{\Gamma }(s+1)}(s\ne -1,-2,-3,\dots )}$$

$${\displaystyle \underset{\mathrm{Re}(\mu )\to \mathrm{\infty }}{lim}{\mathrm{Li}}_{-n}(e^{\mu })=-(-1{)}^n{e}^{-\mu }(n=1,2,3,\dots )}$$

$${\displaystyle \underset{\mathrm{Re}(s)\to \mathrm{\infty }}{lim}{\mathrm{Li}}_s(z)=z}$$

$${\displaystyle \underset{\mathrm{Re}(s)\to -\mathrm{\infty }}{lim}{\mathrm{Li}}_s(e^{\mu })=\mathrm{\Gamma }(1-s)(-\mu {)}^{s-1}(-\pi <\mathrm{Im}(\mu )<\pi )}$$

$${\displaystyle \underset{\mathrm{Re}(s)\to -\mathrm{\infty }}{lim}{\mathrm{Li}}_s(-e^{\mu })=\mathrm{\Gamma }(1-s)\left[(-\mu -i\pi {)}^{s-1}+(-\mu +i\pi {)}^{s-1}\right](\mathrm{Im}(\mu )=0)}$$

Wood's first limit for Re(µ) → ∞ has been corrected in accordance with his equation 11.3. The limit for Re(s) → −∞ follows from the general relation of the polylogarithm with the Hurwitz zeta function (see above).

Dilogarithm

Main article: Spence's function

The dilogarithm is the polylogarithm of order s = 2. An alternate integral expression of the dilogarithm for arbitrary complex argument z is (Abramowitz & Stegun 1972, § 27.7):

$${\displaystyle {\mathrm{Li}}_2(z)=-{\int }_0^z\frac{\ln (1-t)}{t}dt=-{\int }_0^1\frac{\ln (1-zt)}{t}dt.}$$

A source of confusion is that some computer algebra systems define the dilogarithm as dilog(z) = Li₂(1−z).

In the case of real z ≥ 1 the first integral expression for the dilogarithm can be written as

$${\displaystyle {\mathrm{Li}}_2(z)=\frac{{\pi }^2}{6}-{\int }_1^z\frac{\ln (t-1)}{t}dt-i\pi \ln z}$$

from which expanding ln(t−1) and integrating term by term we obtain

$${\displaystyle {\mathrm{Li}}_2(z)=\frac{{\pi }^2}{3}-\frac{1}{2}(\ln z{)}^2-\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2{z}^k}-i\pi \ln z(z\ge 1).}$$

The Abel identity for the dilogarithm is given by (Abel 1881)

$${\displaystyle {\mathrm{Li}}_2\left(\frac{x}{1-y}\right)+{\mathrm{Li}}_2\left(\frac{y}{1-x}\right)-{\mathrm{Li}}_2\left(\frac{xy}{(1-x)(1-y)}\right)={\mathrm{Li}}_2(x)+{\mathrm{Li}}_2(y)+\ln (1-x)\ln (1-y)}$$

$${\displaystyle (\mathrm{Re}(x)\le \frac{1}{2}\wedge \mathrm{Re}(y)\le \frac{1}{2}\vee \mathrm{Im}(x)>0\wedge \mathrm{Im}(y)>0\vee \mathrm{Im}(x)<0\wedge \mathrm{Im}(y)<0\vee \dots ).}$$

This is immediately seen to hold for either x = 0 or y = 0, and for general arguments is then easily verified by differentiation ∂/∂x ∂/∂y. For y = 1−x the identity reduces to Euler's reflection formula

$${\displaystyle {\mathrm{Li}}_2\left(x\right)+{\mathrm{Li}}_2\left(1-x\right)=\frac{1}{6}{\pi }^2-\ln (x)\ln (1-x),}$$

where Li₂(1) = ζ(2) = ¹⁄₆ π² has been used and x may take any complex value.

In terms of the new variables u = x/(1−y), v = y/(1−x) the Abel identity reads

$${\displaystyle {\mathrm{Li}}_2(u)+{\mathrm{Li}}_2(v)-{\mathrm{Li}}_2(uv)={\mathrm{Li}}_2\left(\frac{u-uv}{1-uv}\right)+{\mathrm{Li}}_2\left(\frac{v-uv}{1-uv}\right)+\ln \left(\frac{1-u}{1-uv}\right)\ln \left(\frac{1-v}{1-uv}\right),}$$

which corresponds to the pentagon identity given in (Rogers 1907).

From the Abel identity for x = y = 1−z and the square relationship we have Landen's identity

$${\displaystyle {\mathrm{Li}}_2(1-z)+{\mathrm{Li}}_2\left(1-\frac{1}{z}\right)=-\frac{1}{2}(\ln z{)}^2(z\notin ]-\mathrm{\infty };0]),}$$

and applying the reflection formula to each dilogarithm we find the inversion formula

$${\displaystyle {\mathrm{Li}}_2(z)+{\mathrm{Li}}_2(1/z)=-\frac{1}{6}{\pi }^2-\frac{1}{2}[\ln (-z){]}^2(z\notin [0;1[),}$$

and for real z ≥ 1 also

$${\displaystyle {\mathrm{Li}}_2(z)+{\mathrm{Li}}_2(1/z)=\frac{1}{3}{\pi }^2-\frac{1}{2}(\ln z{)}^2-i\pi \ln z.}$$

Known closed-form evaluations of the dilogarithm at special arguments are collected in the table below. Arguments in the first column are related by reflection x ↔ 1−x or inversion x ↔ ¹⁄_x to either x = 0 or x = −1; arguments in the third column are all interrelated by these operations.

Maximon (2003) discusses the 17th to 19th century references. The reflection formula was already published by Landen in 1760, prior to its appearance in a 1768 book by Euler (Maximon 2003, § 10); an equivalent to Abel's identity was already published by Spence in 1809, before Abel wrote his manuscript in 1826 (Zagier 1989, § 2). The designation bilogarithmische Function was introduced by Carl Johan Danielsson Hill (professor in Lund, Sweden) in 1828 (Maximon 2003, § 10). Don Zagier (1989) has remarked that the dilogarithm is the only mathematical function possessing a sense of humor.

Special values of the dilogarithm

${\displaystyle x}$	${\displaystyle {\mathrm{Li}}_2(x)}$	${\displaystyle x}$	${\displaystyle {\mathrm{Li}}_2(x)}$
${\displaystyle -1}$	${\displaystyle -\frac{1}{12}{\pi }^2}$	${\displaystyle -\varphi }$	${\displaystyle -\frac{1}{10}{\pi }^2-{\ln }^2\varphi }$
${\displaystyle 0}$	${\displaystyle 0}$	${\displaystyle -1/\varphi }$	${\displaystyle -\frac{1}{15}{\pi }^2+\frac{1}{2}{\ln }^2\varphi }$
${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{12}{\pi }^2-\frac{1}{2}{\ln }^{2}2}$	${\displaystyle 1/{\varphi }^2}$	${\displaystyle \frac{1}{15}{\pi }^2-{\ln }^2\varphi }$
${\displaystyle 1}$	${\displaystyle \frac{1}{6}{\pi }^2}$	${\displaystyle 1/\varphi }$	${\displaystyle \frac{1}{10}{\pi }^2-{\ln }^2\varphi }$
${\displaystyle 2}$	${\displaystyle \frac{1}{4}{\pi }^2-\pi i\ln 2}$	${\displaystyle \varphi }$	${\displaystyle \frac{11}{15}{\pi }^2+\frac{1}{2}{\ln }^2(-1/\varphi )}$
		${\displaystyle {\varphi }^2}$	${\displaystyle -\frac{11}{15}{\pi }^2-{\ln }^2(-\varphi )}$

Here ${\displaystyle \varphi =\frac{1}{2}(\sqrt{5}+1)}$ denotes the golden ratio.

Polylogarithm ladders

Leonard Lewin discovered a remarkable and broad generalization of a number of classical relationships on the polylogarithm for special values. These are now called polylogarithm ladders. Define ${\displaystyle \rho =\frac{1}{2}(\sqrt{5}-1)}$ as the reciprocal of the golden ratio. Then two simple examples of dilogarithm ladders are

$${\displaystyle {\mathrm{Li}}_2({\rho }^6)=4{\mathrm{Li}}_2({\rho }^3)+3{\mathrm{Li}}_2({\rho }^2)-6{\mathrm{Li}}_2(\rho )+\frac{7}{30}{\pi }^2}$$

given by Coxeter (1935) and

$${\displaystyle {\mathrm{Li}}_2(\rho )=\frac{1}{10}{\pi }^2-{\ln }^2\rho }$$

given by Landen. Polylogarithm ladders occur naturally and deeply in K-theory and algebraic geometry. Polylogarithm ladders provide the basis for the rapid computations of various mathematical constants by means of the BBP algorithm (Bailey, Borwein & Plouffe 1997).

Monodromy

The polylogarithm has two branch points; one at z = 1 and another at z = 0. The second branch point, at z = 0, is not visible on the main sheet of the polylogarithm; it becomes visible only when the function is analytically continued to its other sheets. The monodromy group for the polylogarithm consists of the homotopy classes of loops that wind around the two branch points. Denoting these two by m₀ and m₁, the monodromy group has the group presentation

$${\displaystyle ⟨m_0,m_1|w=m_0{m}_1{m}_0^{-1}{m}_1^{-1},w{m}_1=m_{1}w⟩.}$$

For the special case of the dilogarithm, one also has that wm₀ = m₀w, and the monodromy group becomes the Heisenberg group (identifying m₀, m₁ and w with x, y, z) (Vepstas 2008).

Antimatter rocket

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Antimatter_rocket 

 

A proposed antimatter rocket

An antimatter rocket is a proposed class of rockets that use antimatter as their power source. There are several designs that attempt to accomplish this goal. The advantage to this class of rocket is that a large fraction of the rest mass of a matter/antimatter mixture may be converted to energy, allowing antimatter rockets to have a far higher energy density and specific impulse than any other proposed class of rocket.

Methods

Antimatter rockets can be divided into three types of application: those that directly use the products of antimatter annihilation for propulsion, those that heat a working fluid or an intermediate material which is then used for propulsion, and those that heat a working fluid or an intermediate material to generate electricity for some form of electric spacecraft propulsion system. The propulsion concepts that employ these mechanisms generally fall into four categories: solid core, gaseous core, plasma core, and beamed core configurations. The alternatives to direct antimatter annihilation propulsion offer the possibility of feasible vehicles with, in some cases, vastly smaller amounts of antimatter but require a lot more matter propellant. Then there are hybrid solutions using antimatter to catalyze fission/fusion reactions for propulsion.

Pure antimatter rocket: direct use of reaction products

Antiproton annihilation reactions produce charged and uncharged pions, in addition to neutrinos and gamma rays. The charged pions can be channelled by a magnetic nozzle, producing thrust. This type of antimatter rocket is a pion rocket or beamed core configuration. It is not perfectly efficient; energy is lost as the rest mass of the charged (22.3%) and uncharged pions (14.38%), lost as the kinetic energy of the uncharged pions (which can't be deflected for thrust); and lost as neutrinos and gamma rays (see antimatter as fuel).

Positron annihilation has also been proposed for rocketry. Annihilation of positrons produces only gamma rays. Early proposals for this type of rocket, such as those developed by Eugen Sänger, assumed the use of some material that could reflect gamma rays, used as a light sail or parabolic shield to derive thrust from the annihilation reaction, but no known form of matter (consisting of atoms or ions) interacts with gamma rays in a manner that would enable specular reflection. The momentum of gamma rays can, however, be partially transferred to matter by Compton scattering.

One method to reach relativistic velocities uses a matter-antimatter GeV gamma ray laser photon rocket made possible by a relativistic proton-antiproton pinch discharge, where the recoil from the laser beam is transmitted by the Mössbauer effect to the spacecraft.

A new annihilation process has allegedly been developed by researchers from Gothenborg University. Several annihilation reactors have been constructed in the past years where hydrogen or deuterium is converted into relativistic particles by laser annihilation. The technology has been demonstrated by research groups led by Prof. Leif Holmlid and Sindre Zeiner-Gundersen at research facilities in both Sweden and Oslo. A third relativistic particle reactor is currently being built at the University of Iceland. The emitted particles from hydrogen annihilation processes may reach 0.94c and can be used in space propulsion. Note, however, that the veracity of Leif Holmlid's research is under dispute.

Thermal antimatter rocket: heating of a propellant

This type of antimatter rocket is termed a thermal antimatter rocket as the energy or heat from the annihilation is harnessed to create an exhaust from non-exotic material or propellant.

The solid core concept uses antiprotons to heat a solid, high-atomic weight (Z), refractory metal core. Propellant is pumped into the hot core and expanded through a nozzle to generate thrust. The performance of this concept is roughly equivalent to that of the nuclear thermal rocket (${\displaystyle I_{\text{sp}}}$ ~ 10³ sec) due to temperature limitations of the solid. However, the antimatter energy conversion and heating efficiencies are typically high due to the short mean path between collisions with core atoms (efficiency ${\displaystyle {\eta }_e}$ ~ 85%). Several methods for the liquid-propellant thermal antimatter engine using the gamma rays produced by antiproton or positron annihilation have been proposed. These methods resemble those proposed for nuclear thermal rockets. One proposed method is to use positron annihilation gamma rays to heat a solid engine core. Hydrogen gas is ducted through this core, heated, and expelled from a rocket nozzle. A second proposed engine type uses positron annihilation within a solid lead pellet or within compressed xenon gas to produce a cloud of hot gas, which heats a surrounding layer of gaseous hydrogen. Direct heating of the hydrogen by gamma rays was considered impractical, due to the difficulty of compressing enough of it within an engine of reasonable size to absorb the gamma rays. A third proposed engine type uses annihilation gamma rays to heat an ablative sail, with the ablated material providing thrust. As with nuclear thermal rockets, the specific impulse achievable by these methods is limited by materials considerations, typically being in the range of 1000–2000 seconds.

The gaseous core system substitutes the low-melting point solid with a high temperature gas (i.e. tungsten gas/plasma), thus permitting higher operational temperatures and performance (${\displaystyle I_{\text{sp}}}$ ~ 2 × 10³ sec). However, the longer mean free path for thermalization and absorption results in much lower energy conversion efficiencies (${\displaystyle {\eta }_e}$ ~ 35%).

The plasma core allows the gas to ionize and operate at even higher effective temperatures. Heat loss is suppressed by magnetic confinement in the reaction chamber and nozzle. Although performance is extremely high (${\displaystyle I_{\text{sp}}}$ ~ 10⁴-10⁵ sec), the long mean free path results in very low energy utilization (${\displaystyle {\eta }_e}$ ~ 10%)

Antimatter power generation

The idea of using antimatter to power an electric space drive has also been proposed. These proposed designs are typically similar to those suggested for nuclear electric rockets. Antimatter annihilations are used to directly or indirectly heat a working fluid, as in a nuclear thermal rocket, but the fluid is used to generate electricity, which is then used to power some form of electric space propulsion system. The resulting system shares many of the characteristics of other charged particle/electric propulsion proposals, that typically being high specific impulse and low thrust (An associated article further detailing antimatter power generation).

Catalyzed fission/fusion or spiked fusion

This is a hybrid approach in which antiprotons are used to catalyze a fission/fusion reaction or to "spike" the propulsion of a fusion rocket or any similar applications.

The antiproton-driven Inertial confinement fusion (ICF) Rocket concept uses pellets for the D-T reaction. The pellet consists of a hemisphere of fissionable material such as U²³⁵ with a hole through which a pulse of antiprotons and positrons is injected. It is surrounded by a hemisphere of fusion fuel, for example deuterium-tritium, or lithium deuteride. Antiproton annihilation occurs at the surface of the hemisphere, which ionizes the fuel. These ions heat the core of the pellet to fusion temperatures.

The antiproton-driven Magnetically Insulated Inertial Confinement Fusion Propulsion (MICF) concept relies on self-generated magnetic field which insulates the plasma from the metallic shell that contains it during the burn. The lifetime of the plasma was estimated to be two orders of magnitude greater than implosion inertial fusion, which corresponds to a longer burn time, and hence, greater gain.

The antimatter-driven P-B¹¹ concept uses antiprotons to ignite the P-B¹¹ reactions in an MICF scheme. Excessive radiation losses are a major obstacle to ignition and require modifying the particle density, and plasma temperature to increase the gain. It was concluded that it is entirely feasible that this system could achieve I_sp~10⁵s.

A different approach was envisioned for AIMStar in which small fusion fuel droplets would be injected into a cloud of antiprotons confined in a very small volume within a reaction Penning trap. Annihilation takes place on the surface of the antiproton cloud, peeling back 0.5% of the cloud. The power density released is roughly comparable to a 1 kJ, 1 ns laser depositing its energy over a 200 μm ICF target.

The ICAN-II project employs the antiproton catalyzed microfission (ACMF) concept which uses pellets with a molar ratio of 9:1 of D-T:U²³⁵ for Nuclear pulse propulsion.

Difficulties with antimatter rockets

The chief practical difficulties with antimatter rockets are the problems of creating antimatter and storing it. Creating antimatter requires input of vast amounts of energy, at least equivalent to the rest energy of the created particle/antiparticle pairs, and typically (for antiproton production) tens of thousands to millions of times more. Most storage schemes proposed for interstellar craft require the production of frozen pellets of antihydrogen. This requires cooling of antiprotons, binding to positrons, and capture of the resulting antihydrogen atoms - tasks which have, as of 2010, been performed only for small numbers of individual atoms. Storage of antimatter is typically done by trapping electrically charged frozen antihydrogen pellets in Penning or Paul traps. There is no theoretical barrier to these tasks being performed on the scale required to fuel an antimatter rocket. However, they are expected to be extremely (and perhaps prohibitively) expensive due to current production abilities being only able to produce small numbers of atoms, a scale approximately 10²³ times smaller than needed for a 10-gram trip to Mars.

Generally, the energy from antiproton annihilation is deposited over such a large region that it cannot efficiently drive nuclear capsules. Antiproton-induced fission and self-generated magnetic fields may greatly enhance energy localization and efficient use of annihilation energy.

A secondary problem is the extraction of useful energy or momentum from the products of antimatter annihilation, which are primarily in the form of extremely energetic ionizing radiation. The antimatter mechanisms proposed to date have for the most part provided plausible mechanisms for harnessing energy from these annihilation products. The classic rocket equation with its "wet" mass (${\displaystyle M_0}$)(with propellant mass fraction) to "dry" mass (${\displaystyle M_1}$)(with payload) fraction (${\displaystyle \frac{M_0}{M_1}}$), the velocity change (${\displaystyle \mathrm{\Delta }v}$) and specific impulse (${\displaystyle I_{\text{sp}}}$) no longer holds due to the mass losses occurring in antimatter annihilation.

Another general problem with high powered propulsion is excess heat or waste heat, and as with antimatter-matter annihilation also includes extreme radiation. A proton-antiproton annihilation propulsion system transforms 39% of the propellant mass into an intense high-energy flux of gamma radiation. The gamma rays and the high-energy charged pions will cause heating and radiation damage if they are not shielded against. Unlike neutrons, they will not cause the exposed material to become radioactive by transmutation of the nuclei. The components needing shielding are the crew, the electronics, the cryogenic tankage, and the magnetic coils for magnetically assisted rockets. Two types of shielding are needed: radiation protection and thermal protection (different from Heat shield or thermal insulation).

Finally, relativistic considerations have to be taken into account. As the by products of annihilation move at relativistic velocities the rest mass changes according to relativistic mass–energy. For example, the total mass–energy content of the neutral pion is converted into gammas, not just its rest mass. It is necessary to use a relativistic rocket equation that takes into account the relativistic effects of both the vehicle and propellant exhaust (charged pions) moving near the speed of light. These two modifications to the two rocket equations result in a mass ratio (${\displaystyle \frac{M_0}{M_1}}$) for a given (${\displaystyle \mathrm{\Delta }v}$) and (${\displaystyle I_{\text{sp}}}$) that is much higher for a relativistic antimatter rocket than for either a classical or relativistic "conventional" rocket.

Modified relativistic rocket equation

The loss of mass specific to antimatter annihilation requires a modification of the relativistic rocket equation given as

${\displaystyle \frac{M_0}{M_1}={\left(\frac{1+\frac{\mathrm{\Delta }v}{c}}{1-\frac{\mathrm{\Delta }v}{c}}\right)}^{\frac{c}{2I_{\text{sp}}}}}$

(I)

where ${\displaystyle c}$ is the speed of light, and ${\displaystyle I_{\text{sp}}}$ is the specific impulse (i.e. ${\displaystyle I_{\text{sp}}}$=0.69${\displaystyle c}$).

The derivative form of the equation is

${\displaystyle \frac{d{M}_{\text{ship}}}{M_{\text{ship}}}=\frac{-dv(1-I_{\text{sp}}\frac{v}{c^2})}{(1-\frac{v^2}{c^2})(-\frac{I_{\text{sp}}{v}^2}{c^2}+(1-a)v+a{I}_{\text{sp}})}}$

(II)

where ${\displaystyle M_{\text{ship}}}$ is the non-relativistic (rest) mass of the rocket ship, and ${\displaystyle a}$ is the fraction of the original (on board) propellant mass (non-relativistic) remaining after annihilation (i.e., ${\displaystyle a}$=0.22 for the charged pions).

Eq.II is difficult to integrate analytically. If it is assumed that ${\displaystyle v\sim I_{\text{sp}}}$, such that ${\displaystyle (1-\frac{I_{\text{sp}}v}{c^2})\sim (1-\frac{v^2}{c^2})}$ then the resulting equation is

${\displaystyle \frac{d{M}_{\text{ship}}}{M_{\text{ship}}}=\frac{-dv}{(-\frac{I_{\text{sp}}{v}^2}{c^2}+(1-a)v+a{I}_{\text{sp}})}}$

(III)

Eq.III can be integrated and the integral evaluated for ${\displaystyle M_0}$ and ${\displaystyle M_1}$, and initial and final velocities (${\displaystyle v_i=0}$ and ${\displaystyle v_f=\mathrm{\Delta }v}$). The resulting relativistic rocket equation with loss of propellant is

${\displaystyle \frac{M_0}{M_1}={\left(\frac{(-2I_{\text{sp}}\mathrm{\Delta }v/c^2+1-a-\sqrt{(1-a{)}^2+4a{I}_{\text{sp}}^2/c^2})(1-a+\sqrt{(1-a{)}^2+4a{I}_{\text{sp}}^2/c^2})}{(-2I_{\text{sp}}\mathrm{\Delta }v/c^2+1-a+\sqrt{(1-a{)}^2+4a{I}_{\text{sp}}^2/c^2})(1-a-\sqrt{(1-a{)}^2+4a{I}_{\text{sp}}^2/c^2})}\right)}^{\frac{1}{\sqrt{(1-a{)}^2+4a{I}_{\text{sp}}^2/c^2}}}}$

(IV)

Other general issues

The cosmic background hard radiation will ionize the rocket's hull over time and poses a health threat. Also, gas plasma interactions may cause space charge. The major interaction of concern is differential charging of various parts of a spacecraft, leading to high electric fields and arcing between spacecraft components. This can be resolved with well placed plasma contactor. However, there is no solution yet for when plasma contactors are turned off to allow maintenance work on the hull. Long term space flight at interstellar velocities causes erosion of the rocket's hull due to collision with particles, gas, dust and micrometeorites. At 0.2${\displaystyle c}$ for a 6 light year distance, erosion is estimated to be in the order of about 30 kg/m² or about 1 cm of aluminum shielding.

Nuclear pulse propulsion

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Nuclear_pulse_propulsion 

An artist's conception of the Project Orion "basic" spacecraft, powered by nuclear pulse propulsion.

Nuclear pulse propulsion or external pulsed plasma propulsion is a hypothetical method of spacecraft propulsion that uses nuclear explosions for thrust. It originated as Project Orion with support from DARPA, after a suggestion by Stanislaw Ulam in 1947. Newer designs using inertial confinement fusion have been the baseline for most later designs, including Project Daedalus and Project Longshot.

History

Los Alamos

Main article: Los Alamos National Laboratory

Calculations for a potential use of this technology were made at the laboratory from and toward the close of the 1940s to the mid-1950s.

Project Orion

Main article: Project Orion (nuclear propulsion)

A nuclear pulse propulsion unit. The explosive charge ablatively vaporizes the propellant, propelling it away from the charge, and simultaneously creating a plasma out of the propellant. The propellant then goes on to impact the pusher plate at the bottom of the Orion spacecraft, imparting a pulse of 'pushing' energy.

Project Orion was the first serious attempt to design a nuclear pulse rocket. A design was formed at General Atomics during the late 1950s and early 1960s, with the idea of reacting small directional nuclear explosives utilizing a variant of the Teller–Ulam two-stage bomb design against a large steel pusher plate attached to the spacecraft with shock absorbers. Efficient directional explosives maximized the momentum transfer, leading to specific impulses in the range of 6,000 seconds, or about thirteen times that of the Space Shuttle main engine. With refinements a theoretical maximum of 100,000 seconds (1 MN·s/kg) might be possible. Thrusts were in the millions of tons, allowing spacecraft larger than 8×10⁶ tons to be built with 1958 materials.

The reference design was to be constructed of steel using submarine-style construction with a crew of more than 200 and a vehicle takeoff weight of several thousand tons. This single-stage reference design would reach Mars and return in four weeks from the Earth's surface (compared to 12 months for NASA's current chemically powered reference mission). The same craft could visit Saturn's moons in a seven-month mission (compared to chemically powered missions of about nine years). Notable engineering problems that occurred were related to crew shielding and pusher-plate lifetime.

Although the system appeared to be workable, the project was shut down in 1965, primarily because the Partial Test Ban Treaty made it illegal; in fact, before the treaty, the US and Soviet Union had already separately detonated a combined number of at least nine nuclear bombs, including thermonuclear, in space, i.e., at altitudes of over 100 km (see high-altitude nuclear explosions). Ethical issues complicated the launch of such a vehicle within the Earth's magnetosphere: calculations using the (disputed) linear no-threshold model of radiation damage showed that the fallout from each takeoff would cause the death of approximately 1 to 10 individuals. In a threshold model, such extremely low levels of thinly distributed radiation would have no associated ill-effects, while under hormesis models, such tiny doses would be negligibly beneficial. The use of less efficient clean nuclear bombs for achieving orbit and then more efficient, higher yield dirtier bombs for travel would significantly reduce the amount of fallout caused from an Earth-based launch.

One useful mission would be to deflect an asteroid or comet on collision course with the Earth, depicted dramatically in the 1998 film Deep Impact. The high performance would permit even a late launch to succeed, and the vehicle could effectively transfer a large amount of kinetic energy to the asteroid by simple impact. The prospect of an imminent asteroid impact would obviate concerns over the few predicted deaths from fallout. An automated mission would remove the challenge of designing a shock absorber that would protect the crew.

Orion is one of very few interstellar space drives that could theoretically be constructed with available technology, as discussed in a 1968 paper, "Interstellar Transport" by Freeman Dyson.

Project Daedalus

Main article: Project Daedalus

Project Daedalus was a study conducted between 1973 and 1978 by the British Interplanetary Society (BIS) to design an interstellar uncrewed spacecraft that could reach a nearby star within about 50 years. A dozen scientists and engineers led by Alan Bond worked on the project. At the time fusion research appeared to be making great strides, and in particular, inertial confinement fusion (ICF) appeared to be adaptable as a rocket engine.

ICF uses small pellets of fusion fuel, typically lithium deuteride (⁶Li²H) with a small deuterium/tritium trigger at the center. The pellets are thrown into a reaction chamber where they are hit on all sides by lasers or another form of beamed energy. The heat generated by the beams explosively compresses the pellet to the point where fusion takes place. The result is a hot plasma, and a very small "explosion" compared to the minimum size bomb that would be required to instead create the necessary amount of fission.

For Daedalus, this process was to be run within a large electromagnet that formed the rocket engine. After the reaction, ignited by electron beams, the magnet funnelled the hot gas to the rear for thrust. Some of the energy was diverted to run the ship's systems and engine. In order to make the system safe and energy efficient, Daedalus was to be powered by a helium-3 fuel collected from Jupiter.

Medusa

Conceptual diagram of a Medusa propulsion spacecraft, showing: (A) the payload capsule, (B) the winch mechanism, (C) the optional main tether cable, (D) riser tethers, and (E) the parachute mechanism.

 

Operating sequence of the Medusa propulsion system. This diagram shows the operating sequence of a Medusa propulsion spacecraft (1) Starting at moment of explosive-pulse unit firing, (2) As the explosive pulse reaches the parachute canopy, (3) Pushes the canopy, accelerating it away from the explosion as the spacecraft plays out the main tether with the winch, generating electricity as it extends, and accelerating the spacecraft, (4) And finally winches the spacecraft forward to the canopy and uses excess electricity for other purposes.

The Medusa design has more in common with solar sails than with conventional rockets. It was envisioned by Johndale Solem in the 1990s and published in the Journal of the British Interplanetary Society (JBIS).

A Medusa spacecraft would deploy a large sail ahead of it, attached by independent cables, and then launch nuclear explosives forward to detonate between itself and its sail. The sail would be accelerated by the plasma and photonic impulse, running out the tethers as when a fish flees a fisher, generating electricity at the "reel". The spacecraft would use some of the generated electricity to reel itself up towards the sail, constantly smoothly accelerating as it goes.

In the original design, multiple tethers connected to multiple motor generators. The advantage over the single tether is to increase the distance between the explosion and the tethers, thus reducing damage to the tethers.

For heavy payloads, performance could be improved by taking advantage of lunar materials, for example, wrapping the explosive with lunar rock or water, stored previously at a stable Lagrange point.

Medusa performs better than the classical Orion design because its sail intercepts more of the explosive impulse, its shock-absorber stroke is much longer, and its major structures are in tension and hence can be quite lightweight. Medusa-type ships would be capable of a specific impulse between 50,000 and 100,000 seconds (500 to 1000 kN·s/kg).

Medusa became widely known to the public in the BBC documentary film To Mars By A-Bomb: The Secret History of Project Orion. A short film shows an artist's conception of how the Medusa spacecraft works "by throwing bombs into a sail that's ahead of it".

Project Longshot

Main article: Project Longshot

Project Longshot was a NASA-sponsored research project carried out in conjunction with the US Naval Academy in the late 1980s. Longshot was in some ways a development of the basic Daedalus concept, in that it used magnetically funneled ICF. The key difference was that they felt that the reaction could not power both the rocket and the other systems, and instead included a 300 kW conventional nuclear reactor for running the ship. The added weight of the reactor reduced performance somewhat, but even using LiD fuel it would be able to reach neighboring star Alpha Centauri in 100 years (approx. velocity of 13,411 km/s, at a distance of 4.5 light years, equivalent to 4.5% of light speed).

Antimatter-catalyzed nuclear reaction

Main article: Antimatter-catalyzed nuclear pulse propulsion

In the mid-1990s, research at Pennsylvania State University led to the concept of using antimatter to catalyze nuclear reactions. Antiprotons would react inside the nucleus of uranium, releasing energy that breaks the nucleus apart as in conventional nuclear reactions. Even a small number of such reactions can start the chain reaction that would otherwise require a much larger volume of fuel to sustain. Whereas the "normal" critical mass for plutonium is about 11.8 kilograms (for a sphere at standard density), with antimatter catalyzed reactions this could be well under one gram.

Several rocket designs using this reaction were proposed, some which would use all-fission reactions for interplanetary missions, and others using fission-fusion (effectively a very small version of Orion's bombs) for interstellar missions.

Magneto-inertial fusion

MSNW magneto-inertial fusion driven rocket

Concept graphic of a fusion-driven rocket powered spacecraft arriving at Mars
Designer	MSNW LLC
Application	Interplanetary
Status	Theoretical
Performance
Specific impulse	1,606 s to 5,722 s (depending on fusion gain)
Burn time	1 day to 90 days (10 days optimal with gain of 40)
Notes	Fuel: Deuterium-tritium cryogenic pellet Propellant: Lithium or aluminum Power requirements: 100 kW to 1,000 kW

NASA funded MSNW LLC and the University of Washington in 2011 to study and develop a fusion rocket through the NASA Innovative Advanced Concepts NIAC Program.

The rocket uses a form of magneto-inertial fusion to produce a direct thrust fusion rocket. Magnetic fields cause large metal rings to collapse around the deuterium-tritium plasma, triggering fusion. The energy heats and ionizes the shell of metal formed by the crushed rings. The hot, ionized metal is shot out of a magnetic rocket nozzle at a high speed (up to 30 km/s). Repeating this process roughly every minute would propel the spacecraft. The fusion reaction is not self-sustaining and requires electrical energy to explode each pulse. With electrical requirements estimated to be between 100 kW to 1,000 kW (300 kW average), designs incorporate solar panels to produce the required energy.

Foil Liner Compression creates fusion at the proper energy scale. The proof of concept experiment in Redmond, Washington, was to use aluminum liners for compression. However, the ultimate design was to use lithium liners.

Performance characteristics are dependent on the fusion energy gain factor achieved by the reactor. Gains were expected to be between 20 and 200, with an estimated average of 40. Higher gains produce higher exhaust velocity, higher specific impulse and lower electrical power requirements. The table below summarizes different performance characteristics for a theoretical 90-day Mars transfer at gains of 20, 40 and 200.

FDR parameters for 90 Mars transfer burn

Total gain	Gain of 20	Gain of 40	Gain of 200
Liner mass (kg)	0.365	0.365	0.365
Specific impulse (s)	1,606	2,435	5,722
Mass fraction	0.33	0.47	0.68
Specific mass (kg/kW)	0.8	0.53	0.23
Mass propellant (kg)	110,000	59,000	20,000
Mass initial (kg)	184,000	130,000	90,000
Electrical power required (kW)	1,019	546	188

By April 2013, MSNW had demonstrated subcomponents of the systems: heating deuterium plasma up to fusion temperatures and concentrating the magnetic fields needed to create fusion. They planned to put the two technologies together for a test before the end of 2013.

Pulsed fission-fusion propulsion

Pulsed Fission-Fusion (PuFF) propulsion is reliant on principles similar to magneto-inertial fusion, It aims to solve the problem of the extreme stress induced on containment by an Orion-like motor by ejecting the plasma obtained from small fuel pellets that undergo autocatalytic fission and fusion reactions initiated by a Z-pinch. It is a theoretical propulsion system researched through the NIAC Program by the University of Alabama in Huntsville. It is in essence a fusion rocket that uses a Z-pinch configuration, but coupled with a fission reaction to boost the fusion process.

A PuFF fuel pellet, around 1 cm in diameter, consists of two components: A deuterium-tritium (D-T) cylinder of plasma, called the target, which undergoes fusion, and a surrounding U-235 sheath that undergoes fission enveloped by a lithium liner. Liquid lithium, serving as a moderator, fills the space between the D-T cylinder and the uranium sheath. Current is run through the liquid lithium, a Lorentz force is generated which then compresses the D-T plasma by a factor of 10 in what is known as a Z-pinch. The compressed plasma reaches criticality and undergoes fusion reactions. However, the fusion energy gain (Q) of these reactions is far below breakeven (Q < 1), meaning that the reaction consumes more energy than it produces.

In a PuFF design, the fast neutrons released by the initial fusion reaction induce fission in the U-235 sheath. The resultant heat causes the sheath to expand, increasing its implosion velocity onto the D-T core and compressing it further, releasing more fast neutrons. Those again amplify the fission rate in the sheath, rendering the process autocatalytic. It is hoped that this results in a complete burn up of both the fission and fusion fuels, making PuFF more efficient than other nuclear pulse concepts. Much like in a magneto-inertial fusion rocket, the performance of the engine is dependent on the degree to which the fusion gain of the D-T target is increased.

One "pulse" consist of the injection of a fuel pellet into the combustion chamber, its consumption through a series of fission-fusion reactions, and finally the ejection of the released plasma through a magnetic nozzle, thus generating thrust. A single pulse is expected to take only a fraction of a second to complete.