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0

1

$begingroup$

If have a function $f$ given by a series
$$
f(z) = sum_{n,m = 0}^infty u_{n,m} z^{n + m t}
$$
for some $tinmathbb{R}^+$.

Is there an straightforward way (something similar to the inversion formula for Taylor series) to derive a similar series for $f^{-1}(z)$?

power-series taylor-expansion inverse inverse-function

share|cite|improve this question

asked Mar 22 at 22:05

Dark MalthorpDark Malthorp

938314

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Lagrange inversion? en.wikipedia.org/wiki/Lagrange_inversion_theorem
  $endgroup$
  – Lord Shark the Unknown
  Mar 22 at 22:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I'm not sure how that would work in this case, since it's not exactly a Taylor series
  $endgroup$
  – Dark Malthorp
  Mar 22 at 22:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is the inversion formula for Taylor series ?
  $endgroup$
  – Yves Daoust
  Mar 22 at 22:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  With $w:=z^t$, you in fact have a bivariate series $f(z,w)=sum u_{n,m}z^nw^m$.
  $endgroup$
  – Yves Daoust
  Mar 22 at 22:14
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  OK. Does that help me invert it?
  $endgroup$
  – Dark Malthorp
  Mar 23 at 1:04
  

  
  
  
  

 | 
show 2 more comments

0

1

$begingroup$

If have a function $f$ given by a series
$$
f(z) = sum_{n,m = 0}^infty u_{n,m} z^{n + m t}
$$
for some $tinmathbb{R}^+$.

Is there an straightforward way (something similar to the inversion formula for Taylor series) to derive a similar series for $f^{-1}(z)$?

power-series taylor-expansion inverse inverse-function

share|cite|improve this question

asked Mar 22 at 22:05

Dark MalthorpDark Malthorp

938314

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Lagrange inversion? en.wikipedia.org/wiki/Lagrange_inversion_theorem
  $endgroup$
  – Lord Shark the Unknown
  Mar 22 at 22:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I'm not sure how that would work in this case, since it's not exactly a Taylor series
  $endgroup$
  – Dark Malthorp
  Mar 22 at 22:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is the inversion formula for Taylor series ?
  $endgroup$
  – Yves Daoust
  Mar 22 at 22:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  With $w:=z^t$, you in fact have a bivariate series $f(z,w)=sum u_{n,m}z^nw^m$.
  $endgroup$
  – Yves Daoust
  Mar 22 at 22:14
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  OK. Does that help me invert it?
  $endgroup$
  – Dark Malthorp
  Mar 23 at 1:04
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

If have a function $f$ given by a series
$$
f(z) = sum_{n,m = 0}^infty u_{n,m} z^{n + m t}
$$
for some $tinmathbb{R}^+$.

Is there an straightforward way (something similar to the inversion formula for Taylor series) to derive a similar series for $f^{-1}(z)$?

power-series taylor-expansion inverse inverse-function

share|cite|improve this question

asked Mar 22 at 22:05

Dark MalthorpDark Malthorp

938314

$endgroup$

share|cite|improve this question

asked Mar 22 at 22:05

Dark MalthorpDark Malthorp

938314

share|cite|improve this question

asked Mar 22 at 22:05

Dark MalthorpDark Malthorp

938314

asked Mar 22 at 22:05

Dark MalthorpDark Malthorp

938314

asked Mar 22 at 22:05

Dark MalthorpDark Malthorp

938314