Solving quadratic equations with continued fractions

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Solving_quadratic_equations_with_continued_fractions

In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is

${\displaystyle a{x}^2+bx+c=0,}$

where a ≠ 0.

The quadratic equation on a number ${\displaystyle x}$ can be solved using the well-known quadratic formula, which can be derived by completing the square. That formula always gives the roots of the quadratic equation, but the solutions are expressed in a form that often involves a quadratic irrational number, which is an algebraic fraction that can be evaluated as a decimal fraction only by applying an additional root extraction algorithm.

If the roots are real, there is an alternative technique that obtains a rational approximation to one of the roots by manipulating the equation directly. The method works in many cases, and long ago it stimulated further development of the analytical theory of continued fractions.

Simple example

Here is a simple example to illustrate the solution of a quadratic equation using continued fractions. We begin with the equation

${\displaystyle x^2=2}$

and manipulate it directly. Subtracting one from both sides we obtain

${\displaystyle x^2-1=1.}$

This is easily factored into

${\displaystyle (x+1)(x-1)=1}$

from which we obtain

${\displaystyle (x-1)=\frac{1}{1+x}}$

and finally

${\displaystyle x=1+\frac{1}{1+x}.}$

Now comes the crucial step. We substitute this expression for x back into itself, recursively, to obtain

${\displaystyle x=1+\frac{1}{1+\left(1+\frac{1}{1+x}\right)}=1+\frac{1}{2+\frac{1}{1+x}}.}$

But now we can make the same recursive substitution again, and again, and again, pushing the unknown quantity x as far down and to the right as we please, and obtaining in the limit the infinite continued fraction

${\displaystyle x=1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\ddots }}}}}=\sqrt{2}.}$

By applying the fundamental recurrence formulas we may easily compute the successive convergents of this continued fraction to be 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, ..., where each successive convergent is formed by taking the numerator plus the denominator of the preceding term as the denominator in the next term, then adding in the preceding denominator to form the new numerator. This sequence of denominators is a particular Lucas sequence known as the Pell numbers.

Algebraic explanation

We can gain further insight into this simple example by considering the successive powers of

${\displaystyle \omega =\sqrt{2}-1.}$

That sequence of successive powers is given by

${\displaystyle \begin{array}{rlrlrl}{\omega }^2& =3-2\sqrt{2},& {\omega }^3& =5\sqrt{2}-7,& {\omega }^4& =17-12\sqrt{2},\\ {\omega }^5& =29\sqrt{2}-41,& {\omega }^6& =99-70\sqrt{2},& {\omega }^7& =169\sqrt{2}-239,\end{array}}$

and so forth. Notice how the fractions derived as successive approximants to √2 appear in this geometric progression.

Since 0 < ω < 1, the sequence {ωⁿ} clearly tends toward zero, by well-known properties of the positive real numbers. This fact can be used to prove, rigorously, that the convergents discussed in the simple example above do in fact converge to √2, in the limit.

We can also find these numerators and denominators appearing in the successive powers of

${\displaystyle {\omega }^{-1}=\sqrt{2}+1.}$

The sequence of successive powers {ω⁻ⁿ} does not approach zero; it grows without limit instead. But it can still be used to obtain the convergents in our simple example.

Notice also that the set obtained by forming all the combinations a + b√2, where a and b are integers, is an example of an object known in abstract algebra as a ring, and more specifically as an integral domain. The number ω is a unit in that integral domain. See also algebraic number field.

General quadratic equation

Continued fractions are most conveniently applied to solve the general quadratic equation expressed in the form of a monic polynomial

${\displaystyle x^2+bx+c=0}$

which can always be obtained by dividing the original equation by its leading coefficient. Starting from this monic equation we see that

${\displaystyle \begin{array}{rl}{x}^2+bx& =-c\\ x+b& =\frac{-c}{x}\\ x& =-b-\frac{c}{x}\end{array}}$

But now we can apply the last equation to itself recursively to obtain

${\displaystyle x=-b-\frac{c}{-b-\frac{c}{-b-\frac{c}{-b-\frac{c}{-b-\ddots }}}}}$

If this infinite continued fraction converges at all, it must converge to one of the roots of the monic polynomial x² + bx + c = 0. Unfortunately, this particular continued fraction does not converge to a finite number in every case. We can easily see that this is so by considering the quadratic formula and a monic polynomial with real coefficients. If the discriminant of such a polynomial is negative, then both roots of the quadratic equation have imaginary parts. In particular, if b and c are real numbers and b² − 4c < 0, all the convergents of this continued fraction "solution" will be real numbers, and they cannot possibly converge to a root of the form u + iv (where v ≠ 0), which does not lie on the real number line.

General theorem

By applying a result obtained by Euler in 1748 it can be shown that the continued fraction solution to the general monic quadratic equation with real coefficients

${\displaystyle x^2+bx+c=0}$

given by

${\displaystyle x=-b-\frac{c}{-b-\frac{c}{-b-\frac{c}{-b-\frac{c}{-b-\ddots }}}}}$

either converges or diverges depending on both the coefficient b and the value of the discriminant, b² − 4c.

If b = 0 the general continued fraction solution is totally divergent; the convergents alternate between 0 and ${\displaystyle \mathrm{\infty }}$. If b ≠ 0 we distinguish three cases.

1. If the discriminant is negative, the fraction diverges by oscillation, which means that its convergents wander around in a regular or even chaotic fashion, never approaching a finite limit.
2. If the discriminant is zero the fraction converges to the single root of multiplicity two.
3. If the discriminant is positive the equation has two real roots, and the continued fraction converges to the larger (in absolute value) of these. The rate of convergence depends on the absolute value of the ratio between the two roots: the farther that ratio is from unity, the more quickly the continued fraction converges.

When the monic quadratic equation with real coefficients is of the form x² = c, the general solution described above is useless because division by zero is not well defined. As long as c is positive, though, it is always possible to transform the equation by subtracting a perfect square from both sides and proceeding along the lines illustrated with √2 above. In symbols, if

${\displaystyle x^2=c(c>0)}$

just choose some positive real number p such that

${\displaystyle p^2<c.}$

Then by direct manipulation we obtain

${\displaystyle \begin{array}{rl}{x}^2-p^2& =c-p^2\\ (x+p)(x-p)& =c-p^2\\ x-p& =\frac{c-p^2}{p+x}\\ x& =p+\frac{c-p^2}{p+x}\\ & =p+\frac{c-p^2}{p+\left(p+\frac{c-p^2}{p+x}\right)}& =p+\frac{c-p^2}{2p+\frac{c-p^2}{2p+\frac{c-p^2}{2p+\ddots }}}\end{array}}$

and this transformed continued fraction must converge because all the partial numerators and partial denominators are positive real numbers.

Complex coefficients

By the fundamental theorem of algebra, if the monic polynomial equation x² + bx + c = 0 has complex coefficients, it must have two (not necessarily distinct) complex roots. Unfortunately, the discriminant b² − 4c is not as useful in this situation, because it may be a complex number. Still, a modified version of the general theorem can be proved.

The continued fraction solution to the general monic quadratic equation with complex coefficients

${\displaystyle x^2+bx+c=0(b\ne 0)}$

given by

${\displaystyle x=-b-\frac{c}{-b-\frac{c}{-b-\frac{c}{-b-\frac{c}{-b-\ddots }}}}}$

converges or not depending on the value of the discriminant, b² − 4c, and on the relative magnitude of its two roots.

Denoting the two roots by r₁ and r₂ we distinguish three cases.

1. If the discriminant is zero the fraction converges to the single root of multiplicity two.
2. If the discriminant is not zero, and |r₁| ≠ |r₂|, the continued fraction converges to the root of maximum modulus (i.e., to the root with the greater absolute value).
3. If the discriminant is not zero, and |r₁| = |r₂|, the continued fraction diverges by oscillation.

In case 2, the rate of convergence depends on the absolute value of the ratio between the two roots: the farther that ratio is from unity, the more quickly the continued fraction converges.

This general solution of monic quadratic equations with complex coefficients is usually not very useful for obtaining rational approximations to the roots, because the criteria are circular (that is, the relative magnitudes of the two roots must be known before we can conclude that the fraction converges, in most cases). But this solution does find useful applications in the further analysis of the convergence problem for continued fractions with complex elements.

Infinite compositions of analytic functions

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Infinite_compositions_of_analytic_functions

In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system.

Although the title of this article specifies analytic functions, there are results for more general functions of a complex variable as well.

Notation

There are several notations describing infinite compositions, including the following:

Forward compositions: ${\displaystyle F_{k,n}(z)=f_k\circ f_{k+1}\circ \cdots \circ f_{n-1}\circ f_n(z).}$

Backward compositions: ${\displaystyle G_{k,n}(z)=f_n\circ f_{n-1}\circ \cdots \circ f_{k+1}\circ f_k(z).}$

In each case convergence is interpreted as the existence of the following limits:

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{F}_{1,n}(z),\underset{n\to \mathrm{\infty }}{lim}{G}_{1,n}(z).}$

For convenience, set F_n(z) = F_1,n(z) and G_n(z) = G_1,n(z).

One may also write ${\displaystyle F_n(z)=\underset{k=1}{\stackrel{n}{R}}{f}_k(z)=f_1\circ f_2\circ \cdots \circ f_n(z)}$ and ${\displaystyle G_n(z)=\underset{k=1}{\stackrel{n}{L}}{g}_k(z)=g_n\circ g_{n-1}\circ \cdots \circ g_1(z)}$

Contraction theorem

Many results can be considered extensions of the following result:

Contraction Theorem for Analytic Functions — Let f be analytic in a simply-connected region S and continuous on the closure S of S. Suppose f(S) is a bounded set contained in S. Then for all z in S there exists an attractive fixed point α of f in S such that:

$${\displaystyle F_n(z)=(f\circ f\circ \cdots \circ f)(z)\to \alpha .}$$

Infinite compositions of contractive functions

Let {f_n} be a sequence of functions analytic on a simply-connected domain S. Suppose there exists a compact set Ω ⊂ S such that for each n, f_n(S) ⊂ Ω.

Forward (inner or right) Compositions Theorem — {F_n} converges uniformly on compact subsets of S to a constant function F(z) = λ.

Backward (outer or left) Compositions Theorem — {G_n} converges uniformly on compact subsets of S to γ ∈ Ω if and only if the sequence of fixed points {γ_n} of the {f_n} converges to γ.

Additional theory resulting from investigations based on these two theorems, particularly Forward Compositions Theorem, include location analysis for the limits obtained in the following reference. For a different approach to Backward Compositions Theorem, see the following reference.

Regarding Backward Compositions Theorem, the example f_2n(z) = 1/2 and f_2n−1(z) = −1/2 for S = {z : |z| < 1} demonstrates the inadequacy of simply requiring contraction into a compact subset, like Forward Compositions Theorem.

For functions not necessarily analytic the Lipschitz condition suffices:

Theorem — Suppose ${\displaystyle S}$ is a simply connected compact subset of ${\displaystyle \mathbb{C}}$ and let ${\displaystyle t_n:S\to S}$ be a family of functions that satisfies

$${\displaystyle \mathrm{\forall }n,\mathrm{\forall }{z}_1,z_2\in S,\mathrm{\exists }\rho :\left|t_n(z_1)-t_n(z_2)\right|\le \rho |z_1-z_2|,\rho <1.}$$

Define:

$${\displaystyle \begin{array}{rl}{G}_n(z)& =\left(t_n\circ t_{n-1}\circ \cdots \circ t_1\right)(z)\\ F_n(z)& =\left(t_1\circ t_2\circ \cdots \circ t_n\right)(z)\end{array}}$$

Then ${\displaystyle F_n(z)\to \beta \in S}$ uniformly on ${\displaystyle S.}$ If ${\displaystyle {\alpha }_n}$ is the unique fixed point of ${\displaystyle t_n}$ then ${\displaystyle G_n(z)\to \alpha }$ uniformly on ${\displaystyle S}$ if and only if ${\displaystyle |{\alpha }_n-\alpha |={\epsilon }_n\to 0}$.

Infinite compositions of other functions

Non-contractive complex functions

Results involving entire functions include the following, as examples. Set

${\displaystyle \begin{array}{rl}{f}_n(z)& =a_{n}z+c_{n,2}{z}^2+c_{n,3}{z}^3+\cdots \\ {\rho }_n& =\underset{r}{sup}\left\{{\left|c_{n,r}\right|}^{\frac{1}{r-1}}\right\}\end{array}}$

Then the following results hold:

Theorem E1 — If a_n ≡ 1,

$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{\rho }_n<\mathrm{\infty }}$$

then F_n → F is entire.

Theorem E2 — Set ε_n = |a_n−1| suppose there exists non-negative δ_n, M₁, M₂, R such that the following holds:

$${\displaystyle \begin{array}{rl}\sum _{n=1}^{\mathrm{\infty }}{\epsilon }_n& <\mathrm{\infty },\\ \sum _{n=1}^{\mathrm{\infty }}{\delta }_n& <\mathrm{\infty },\\ \prod _{n=1}^{\mathrm{\infty }}(1+{\delta }_n)& <M_1,\\ \prod _{n=1}^{\mathrm{\infty }}(1+{\epsilon }_n)& <M_2,\\ {\rho }_n& <\frac{{\delta }_n}{R{M}_1{M}_2}.\end{array}}$$

Then G_n(z) → G(z) is analytic for |z| < R. Convergence is uniform on compact subsets of {z : |z| < R}.

Additional elementary results include:

Theorem GF3 — Suppose ${\displaystyle f_k(z)=z+{\rho }_k{\phi }_k(z)}$ where there exist ${\displaystyle R,M>0}$ such that ${\displaystyle |z|<R}$ implies ${\displaystyle |{\phi }_k(z)|<M,\mathrm{\forall }k,}$ Furthermore, suppose ${\rho }_k\ge 0,\sum _{k=1}^{\mathrm{\infty }}{\rho }_k<\mathrm{\infty }$ and $R>M\sum _{k=1}^{\mathrm{\infty }}{\rho }_k.$ Then for $R\ast <R-M\sum _{k=1}^{\mathrm{\infty }}{\rho }_k$

$${\displaystyle G_n(z)\equiv \left(f_n\circ f_{n-1}\circ \cdots \circ f_1\right)(z)\to G(z)\text{ for }\{z:|z|<R\ast \}}$$

Theorem GF4 — Suppose ${\displaystyle f_k(z)=z+{\rho }_k{\phi }_k(z)}$ where there exist ${\displaystyle R,M>0}$ such that ${\displaystyle |z|<R}$ and ${\displaystyle |\zeta |<R}$ implies ${\displaystyle |{\phi }_k(z)|<M}$ and ${\displaystyle |{\phi }_k(z)-{\phi }_k(\zeta )|\le r|z-\zeta |,\mathrm{\forall }k.}$ Furthermore, suppose ${\rho }_k\ge 0,\sum _{k=1}^{\mathrm{\infty }}{\rho }_k<\mathrm{\infty }$ and $R>M\sum _{k=1}^{\mathrm{\infty }}{\rho }_k.$ Then for $R\ast <R-M\sum _{k=1}^{\mathrm{\infty }}{\rho }_k$

$${\displaystyle F_n(z)\equiv \left(f_1\circ f_2\circ \cdots \circ f_n\right)(z)\to F(z)\text{ for }\{z:|z|<R\ast \}}$$

Example GF1: ${\displaystyle F_{40}(x+iy)=\underset{k=1}{\stackrel{40}{R}}\left(\frac{x+iy}{1+\frac{1}{4^k}(x\cos (y)+iy\sin (x))}\right),[-20,20]}$

Example GF1:Reproductive universe – A topographical (moduli) image of an infinite composition.

Example GF2: ${\displaystyle G_{40}(x+iy)=\underset{k=1}{\stackrel{40}{L}}\left(\frac{x+iy}{1+\frac{1}{2^k}(x\cos (y)+iy\sin (x))}\right),[-20,20]}$

Example GF2:Metropolis at 30K – A topographical (moduli) image of an infinite composition.

Linear fractional transformations

Results for compositions of linear fractional (Möbius) transformations include the following, as examples:

Theorem LFT1 — On the set of convergence of a sequence {F_n} of non-singular LFTs, the limit function is either:

1. a non-singular LFT,
2. a function taking on two distinct values, or
3. a constant.

In (a), the sequence converges everywhere in the extended plane. In (b), the sequence converges either everywhere, and to the same value everywhere except at one point, or it converges at only two points. Case (c) can occur with every possible set of convergence.

Theorem LFT2 — If {F_n} converges to an LFT, then f_n converge to the identity function f(z) = z.

Theorem LFT3 — If f_n → f and all functions are hyperbolic or loxodromic Möbius transformations, then F_n(z) → λ, a constant, for all $z\ne \beta =\underset{n\to \mathrm{\infty }}{lim}{\beta }_n$, where {β_n} are the repulsive fixed points of the {f_n}.

Theorem LFT4 — If f_n → f where f is parabolic with fixed point γ. Let the fixed-points of the {f_n} be {γ_n} and {β_n}. If

$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\left|{\gamma }_n-{\beta }_n\right|<\mathrm{\infty }\text{and}\sum _{n=1}^{\mathrm{\infty }}n\left|{\beta }_{n+1}-{\beta }_n\right|<\mathrm{\infty }}$$

then F_n(z) → λ, a constant in the extended complex plane, for all z.

Examples and applications

Continued fractions

The value of the infinite continued fraction

${\displaystyle \frac{a_1}{b_1+\frac{a_2}{b_2+\cdots }}}$

may be expressed as the limit of the sequence {F_n(0)} where

${\displaystyle f_n(z)=\frac{a_n}{b_n+z}.}$

As a simple example, a well-known result (Worpitsky Circle*) follows from an application of Theorem (A):

Consider the continued fraction

${\displaystyle \frac{a_1\zeta }{1+\frac{a_2\zeta }{1+\cdots }}}$

with

${\displaystyle f_n(z)=\frac{a_n\zeta }{1+z}.}$

Stipulate that |ζ| < 1 and |z| < R < 1. Then for 0 < r < 1,

${\displaystyle |a_n|<rR(1-R)\Rightarrow \left|f_n(z)\right|<rR<R\Rightarrow \frac{a_1\zeta }{1+\frac{a_2\zeta }{1+\cdots }}=F(\zeta )}$, analytic for |z| < 1. Set R = 1/2.

Example. ${\displaystyle F(z)=\frac{(i-1)z}{1+i+z+}\frac{(2-i)z}{1+2i+z+}\frac{(3-i)z}{1+3i+z+}\cdots ,}$ ${\displaystyle [-15,15]}$

Example: Continued fraction1 – Topographical (moduli) image of a continued fraction (one for each point) in the complex plane. [−15,15]

Example. A fixed-point continued fraction form (a single variable).

${\displaystyle f_{k,n}(z)=\frac{{\alpha }_{k,n}{\beta }_{k,n}}{{\alpha }_{k,n}+{\beta }_{k,n}-z},{\alpha }_{k,n}={\alpha }_{k,n}(z),{\beta }_{k,n}={\beta }_{k,n}(z),F_n(z)=\left(f_{1,n}\circ \cdots \circ f_{n,n}\right)(z)}$

${\displaystyle {\alpha }_{k,n}=x\cos (ty)+iy\sin (tx),{\beta }_{k,n}=\cos (ty)+i\sin (tx),t=k/n}$

Example: Infinite Brooch - Topographical (moduli) image of a continued fraction form in the complex plane. (6<x<9.6),(4.8<y<8)

Direct functional expansion

Examples illustrating the conversion of a function directly into a composition follow:

Example 1. Suppose ${\displaystyle \varphi }$ is an entire function satisfying the following conditions:

${\displaystyle \{\begin{array}{ll}\varphi (tz)=t\left(\varphi (z)+\varphi (z{)}^2\right)& |t|>1\\ \varphi (0)=0\\ {\varphi }^{\prime }(0)=1\end{array}}$

Then

${\displaystyle f_n(z)=z+\frac{z^2}{t^n}⟹F_n(z)\to \varphi (z)}$.

Example 2.

${\displaystyle f_n(z)=z+\frac{z^2}{2^n}⟹F_n(z)\to \frac{1}{2}\left(e^{2z}-1\right)}$

Example 3.

${\displaystyle f_n(z)=\frac{z}{1-\frac{z^2}{4^n}}⟹F_n(z)\to \tan (z)}$

Example 4.

${\displaystyle g_n(z)=\frac{2\cdot 4^n}{z}\left(\sqrt{1+\frac{z^2}{4^n}}-1\right)⟹G_n(z)\to \arctan (z)}$

Calculation of fixed-points

Theorem (B) can be applied to determine the fixed-points of functions defined by infinite expansions or certain integrals. The following examples illustrate the process:

Example FP1. For |ζ| ≤ 1 let

${\displaystyle G(\zeta )=\frac{\frac{e^{\zeta }}{4}}{3+\zeta +\frac{\frac{e^{\zeta }}{8}}{3+\zeta +\frac{\frac{e^{\zeta }}{12}}{3+\zeta +\cdots }}}}$

To find α = G(α), first we define:

${\displaystyle \begin{array}{rl}{t}_n(z)& =\frac{\frac{e^{\zeta }}{4n}}{3+\zeta +z}\\ f_n(\zeta )& =t_1\circ t_2\circ \cdots \circ t_n(0)\end{array}}$

Then calculate ${\displaystyle G_n(\zeta )=f_n\circ \cdots \circ f_1(\zeta )}$ with ζ = 1, which gives: α = 0.087118118... to ten decimal places after ten iterations.

Theorem FP2 — Let φ(ζ, t) be analytic in S = {z : |z| < R} for all t in [0, 1] and continuous in t. Set

$${\displaystyle f_n(\zeta )=\frac{1}{n}\sum _{k=1}^n\phi \left(\zeta ,\frac{k}{n}\right).}$$

If |φ(ζ, t)| ≤ r < R for ζ ∈ S and t ∈ [0, 1], then

$${\displaystyle \zeta ={\int }_0^1\phi (\zeta ,t)dt}$$

has a unique solution, α in S, with ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{G}_n(\zeta )=\alpha .}$

Evolution functions

Consider a time interval, normalized to I = [0, 1]. ICAFs can be constructed to describe continuous motion of a point, z, over the interval, but in such a way that at each "instant" the motion is virtually zero (see Zeno's Arrow): For the interval divided into n equal subintervals, 1 ≤ k ≤ n set ${\displaystyle g_{k,n}(z)=z+{\phi }_{k,n}(z)}$ analytic or simply continuous – in a domain S, such that

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{\phi }_{k,n}(z)=0}$ for all k and all z in S,

and ${\displaystyle g_{k,n}(z)\in S}$.

Principal example

${\displaystyle \begin{array}{rl}{g}_{k,n}(z)& =z+\frac{1}{n}\varphi \left(z,\frac{k}{n}\right)\\ G_{k,n}(z)& =\left(g_{k,n}\circ g_{k-1,n}\circ \cdots \circ g_{1,n}\right)(z)\\ G_n(z)& =G_{n,n}(z)\end{array}}$

implies

${\displaystyle {\lambda }_n(z)\doteq G_n(z)-z=\frac{1}{n}\sum _{k=1}^n\varphi \left(G_{k-1,n}(z)\frac{k}{n}\right)\doteq \frac{1}{n}\sum _{k=1}^n\psi \left(z,\frac{k}{n}\right)\sim {\int }_0^1\psi (z,t)dt,}$

where the integral is well-defined if ${\displaystyle \frac{dz}{dt}=\varphi (z,t)}$ has a closed-form solution z(t). Then

${\displaystyle {\lambda }_n(z_0)\approx {\int }_0^1\varphi (z(t),t)dt.}$

Otherwise, the integrand is poorly defined although the value of the integral is easily computed. In this case one might call the integral a "virtual" integral.

Example. ${\displaystyle \varphi (z,t)=\frac{2t-\cos y}{1-\sin x\cos y}+i\frac{1-2t\sin x}{1-\sin x\cos y},{\int }_0^1\psi (z,t)dt}$

Example 1: Virtual tunnels – Topographical (moduli) image of virtual integrals (one for each point) in the complex plane. [−10,10]

Two contours flowing towards an attractive fixed point (red on the left). The white contour (c = 2) terminates before reaching the fixed point. The second contour (c(n) = square root of n) terminates at the fixed point. For both contours, n = 10,000

Example. Let:

${\displaystyle g_n(z)=z+\frac{c_n}{n}\varphi (z),\text{with}f(z)=z+\varphi (z).}$

Next, set ${\displaystyle T_{1,n}(z)=g_n(z),T_{k,n}(z)=g_n(T_{k-1,n}(z)),}$ and T_n(z) = T_n,n(z). Let

${\displaystyle T(z)=\underset{n\to \mathrm{\infty }}{lim}{T}_n(z)}$

when that limit exists. The sequence {T_n(z)} defines contours γ = γ(c_n, z) that follow the flow of the vector field f(z). If there exists an attractive fixed point α, meaning |f(z) − α| ≤ ρ|z − α| for 0 ≤ ρ < 1, then T_n(z) → T(z) ≡ α along γ = γ(c_n, z), provided (for example) ${\displaystyle c_n=\sqrt{n}}$. If c_n ≡ c > 0, then T_n(z) → T(z), a point on the contour γ = γ(c, z). It is easily seen that

${\displaystyle {\oint }_{\gamma }\varphi (\zeta )d\zeta =\underset{n\to \mathrm{\infty }}{lim}\frac{c}{n}\sum _{k=1}^n{\varphi }^2\left(T_{k-1,n}(z)\right)}$

and

${\displaystyle L(\gamma (z))=\underset{n\to \mathrm{\infty }}{lim}\frac{c}{n}\sum _{k=1}^n\left|\varphi \left(T_{k-1,n}(z)\right)\right|,}$

when these limits exist.

These concepts are marginally related to active contour theory in image processing, and are simple generalizations of the Euler method

Self-replicating expansions

Series

The series defined recursively by f_n(z) = z + g_n(z) have the property that the nth term is predicated on the sum of the first n − 1 terms. In order to employ theorem (GF3) it is necessary to show boundedness in the following sense: If each f_n is defined for |z| < M then |G_n(z)| < M must follow before |f_n(z) − z| = |g_n(z)| ≤ Cβ_n is defined for iterative purposes. This is because ${\displaystyle g_n(G_{n-1}(z))}$ occurs throughout the expansion. The restriction

${\displaystyle |z|<R=M-C\sum _{k=1}^{\mathrm{\infty }}{\beta }_k>0}$

serves this purpose. Then G_n(z) → G(z) uniformly on the restricted domain.

Example (S1). Set

${\displaystyle f_n(z)=z+\frac{1}{\rho n^2}\sqrt{z},\rho >\sqrt{\frac{\pi }{6}}}$

and M = ρ². Then R = ρ² − (π/6) > 0. Then, if ${\displaystyle S=\left\{z:|z|<R,\mathrm{Re}(z)>0\right\}}$, z in S implies |G_n(z)| < M and theorem (GF3) applies, so that

${\displaystyle \begin{array}{rl}{G}_n(z)& =z+g_1(z)+g_2(G_1(z))+g_3(G_2(z))+\cdots +g_n(G_{n-1}(z))\\ & =z+\frac{1}{\rho \cdot 1^2}\sqrt{z}+\frac{1}{\rho \cdot 2^2}\sqrt{G_1(z)}+\frac{1}{\rho \cdot 3^2}\sqrt{G_2(z)}+\cdots +\frac{1}{\rho \cdot n^2}\sqrt{G_{n-1}(z)}\end{array}}$

converges absolutely, hence is convergent.

Example (S2): ${\displaystyle f_n(z)=z+\frac{1}{n^2}\cdot \phi (z),\phi (z)=2\cos (x/y)+i2\sin (x/y),>G_n(z)=f_n\circ f_{n-1}\circ \cdots \circ f_1(z),[-10,10],n=50}$

Example (S2)- A topographical (moduli) image of a self generating series.

Products

The product defined recursively by

${\displaystyle f_n(z)=z(1+g_n(z)),|z|⩽M,}$

has the appearance

${\displaystyle G_n(z)=z\prod _{k=1}^n\left(1+g_k\left(G_{k-1}(z)\right)\right).}$

In order to apply Theorem GF3 it is required that:

${\displaystyle \left|z{g}_n(z)\right|\le C{\beta }_n,\sum _{k=1}^{\mathrm{\infty }}{\beta }_k<\mathrm{\infty }.}$

Once again, a boundedness condition must support

${\displaystyle \left|G_{n-1}(z)g_n(G_{n-1}(z))\right|\le C{\beta }_n.}$

If one knows Cβ_n in advance, the following will suffice:

${\displaystyle |z|⩽R=\frac{M}{P}\text{where}P=\prod _{n=1}^{\mathrm{\infty }}\left(1+C{\beta }_n\right).}$

Then G_n(z) → G(z) uniformly on the restricted domain.

Example (P1). Suppose ${\displaystyle f_n(z)=z(1+g_n(z))}$ with ${\displaystyle g_n(z)=\frac{z^2}{n^3},}$ observing after a few preliminary computations, that |z| ≤ 1/4 implies |G_n(z)| < 0.27. Then

${\displaystyle \left|G_n(z)\frac{G_n(z{)}^2}{n^3}\right|<(0.02)\frac{1}{n^3}=C{\beta }_n}$

and

${\displaystyle G_n(z)=z\prod _{k=1}^{n-1}\left(1+\frac{G_k(z{)}^2}{n^3}\right)}$

converges uniformly.

Example (P2).

${\displaystyle g_{k,n}(z)=z\left(1+\frac{1}{n}\phi \left(z,\frac{k}{n}\right)\right),}$

${\displaystyle G_{n,n}(z)=\left(g_{n,n}\circ g_{n-1,n}\circ \cdots \circ g_{1,n}\right)(z)=z\prod _{k=1}^n(1+P_{k,n}(z)),}$

${\displaystyle P_{k,n}(z)=\frac{1}{n}\phi \left(G_{k-1,n}(z),\frac{k}{n}\right),}$

${\displaystyle \prod _{k=1}^{n-1}\left(1+P_{k,n}(z)\right)=1+P_{1,n}(z)+P_{2,n}(z)+\cdots +P_{k-1,n}(z)+R_n(z)\sim {\int }_0^1\pi (z,t)dt+1+R_n(z),}$

${\displaystyle \phi (z)=x\cos (y)+iy\sin (x),{\int }_0^1(z\pi (z,t)-1)dt,[-15,15]:}$

Example (P2): Picasso's Universe – a derived virtual integral from a self-generating infinite product. Click on image for higher resolution.

Continued fractions

Example (CF1): A self-generating continued fraction.

${\displaystyle \begin{array}{rl}{F}_n(z)& =\frac{\rho (z)}{{\delta }_1+}\frac{\rho (F_1(z))}{{\delta }_2+}\frac{\rho (F_2(z))}{{\delta }_3+}\cdots \frac{\rho (F_{n-1}(z))}{{\delta }_n},\\ \rho (z)& =\frac{\cos (y)}{\cos (y)+\sin (x)}+i\frac{\sin (x)}{\cos (y)+\sin (x)},[0<x<20],[0<y<20],{\delta }_k\equiv 1\end{array}}$

Example CF1: Diminishing returns – a topographical (moduli) image of a self-generating continued fraction.

Example (CF2): Best described as a self-generating reverse Euler continued fraction.

${\displaystyle G_n(z)=\frac{\rho (G_{n-1}(z))}{1+\rho (G_{n-1}(z))-}\frac{\rho (G_{n-2}(z))}{1+\rho (G_{n-2}(z))-}\cdots \frac{\rho (G_1(z))}{1+\rho (G_1(z))-}\frac{\rho (z)}{1+\rho (z)-z},}$

${\displaystyle \rho (z)=\rho (x+iy)=x\cos (y)+iy\sin (x),[-15,15],n=30}$

Example CF2: Dream of Gold – a topographical (moduli) image of a self-generating reverse Euler continued fraction.