Series Inversion Lagrange Theorem

The Lagrange theorem on the inversion of series allows us to explicitly write the inverse function to a given analytic function in the form of an infinite series. The theorem has applications in combinatorics.

Content

Wording

Let the function $f(z)$ ${\ displaystyle f (z)}$ analytic at the point $z_{0}$ ${\ displaystyle z_ {0}}$ and $f'(z_{0})\neq 0$ ${\ displaystyle f '(z_ {0}) \ neq 0}$ . Then in some neighborhood of the point $w_{0}=f(z_{0})$ ${\ displaystyle w_ {0} = f (z_ {0})}$ function inverse to it $f^{-1}(w)$ ${\ displaystyle f ^ {- 1} (w)}$ representable by a number of species

f^{-1}(w)=z_{0}+\sum _{n=1}^{\infty }{\frac {1}{n!}}\left.\left({\frac {d^{n-1}}{dz^{n-1}}}\left({\frac {z-z_{0}}{f(z)-w_{0}}}\right)^{n}\right)\right|_{z=z_{0}}(w-w_{0})^{n}.

{\ displaystyle f ^ {- 1} (w) = z_ {0} + \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n!}} \ left. \ left ({\ frac {d ^ {n-1}} {dz ^ {n-1}}} \ left ({\ frac {z-z_ {0}} {f (z) -w_ {0}}} \ right) ^ {n} \ right) \ right | _ {z = z_ {0}} (w-w_ {0}) ^ {n}.}

Applications

Burman Series - Lagrange

The Burman – Lagrange series is defined as the expansion of a holomorphic function $f(z)$ ${\ displaystyle f (z)}$ in powers of another holomorphic function $w(z)$ ${\ displaystyle w (z)}$ and is a generalization of the Taylor series .

Let be $f(z)$ ${\ displaystyle f (z)}$ and $w(z)$ ${\ displaystyle w (z)}$ holomorphic in a neighborhood of some point $a\in \mathbb {C}$ ${\ displaystyle a \ in \ mathbb {C}}$ besides $w(a)=0$ ${\ displaystyle w (a) = 0}$ and $a$ ${\ displaystyle a}$ - simple zero function $w(z)$ ${\ displaystyle w (z)}$ . Now select some area $D\ni a$ ${\ displaystyle D \ ni a}$ , wherein $f$ ${\ displaystyle f}$ and $w$ ${\ displaystyle w}$ holomorphic, and $w$ ${\ displaystyle w}$ univalent in ${\overline {D}}$ ${\ displaystyle {\ overline {D}}}$ . Then there is a decomposition of the form:

f(z)=\sum _{n=0}^{\infty }d_{n}w^{n}(z),

{\ displaystyle f (z) = \ sum _ {n = 0} ^ {\ infty} d_ {n} w ^ {n} (z),}

where are the coefficients $d_{n}$ ${\ displaystyle d_ {n}}$ are calculated by the following expression:

d_{n}={\frac {1}{2\pi i}}\int \limits _{\partial D}{\frac {f(\zeta )w'(\zeta )}{w^{n+1}(\zeta )}}\,d\zeta ={\frac {1}{n!}}\lim _{z\to a}{\frac {d^{n-1}}{dz^{n-1}}}\left\{f'(z){\frac {(z-a)^{n}}{w^{n}(z)}}\right\}.

{\ displaystyle d_ {n} = {\ frac {1} {2 \ pi i}} \ int \ limits _ {\ partial D} {\ frac {f (\ zeta) w '(\ zeta)} {w ^ {n + 1} (\ zeta)}} \, d \ zeta = {\ frac {1} {n!}} \ lim _ {z \ to a} {\ frac {d ^ {n-1}} { dz ^ {n-1}}} \ left \ {f '(z) {\ frac {(za) ^ {n}} {w ^ {n} (z)}} \ right \}.}

Series inversion theorem

A special case of the application of series is the so-called problem of inverting a Taylor series.

Consider a decomposition of the form $w=\sum _{n=1}^{\infty }a_{n}z^{n}$ ${\ displaystyle w = \ sum _ {n = 1} ^ {\ infty} a_ {n} z ^ {n}}$ . We try using the obtained expression to calculate the coefficients of the series $z=\sum _{n=1}^{\infty }b_{n}w^{n}$ ${\ displaystyle z = \ sum _ {n = 1} ^ {\ infty} b_ {n} w ^ {n}}$ :

b_{n}={\frac {1}{n!}}\lim _{z\to 0}{\frac {d^{n-1}}{dz^{n-1}}}\left({\frac {z}{w}}\right)^{n}.

{\ displaystyle b_ {n} = {\ frac {1} {n!}} \ lim _ {z \ to 0} {\ frac {d ^ {n-1}} {dz ^ {n-1}}} \ left ({\ frac {z} {w}} \ right) ^ {n}.}

Generalizations

Under the conditions of the theorem for a superposition of the form $F\circ f^{-1}$ ${\ displaystyle F \ circ f ^ {- 1}}$ fair presentation in the form of a series

F(f^{-1}(w))=z_{0}+\sum _{n=1}^{\infty }{\frac {1}{n!}}\left.\left({\frac {d^{n-1}}{dz^{n-1}}}\left(F'(z)\left({\frac {z-z_{0}}{f(z)-w_{0}}}\right)^{n}\right)\right)\right|_{z=z_{0}}(w-w_{0})^{n}.

{\ displaystyle F (f ^ {- 1} (w)) = z_ {0} + \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n!}} \ left. \ left ({\ frac {d ^ {n-1}} {dz ^ {n-1}}} \ left (F '(z) \ left ({\ frac {z-z_ {0}} {f (z) -w_ {0}}} \ right) ^ {n} \ right) \ right) \ right | _ {z = z_ {0}} (w-w_ {0}) ^ {n}.}

Literature

Shabat B.V. Introduction to complex analysis. - M .: Nauka , 1969 .-- 577 p.

Links

Weisstein, Eric W. Lagrange expansion on Wolfram MathWorld .
Weisstein, Eric W. Lagrange Inversion Theorem on the Wolfram MathWorld website.
Weisstein, Eric W. Bürmann's Theorem on the Wolfram MathWorld website.
Weisstein, Eric W. Series Reversion on Wolfram MathWorld .
Bürmann-Lagrange series (English)