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The coefficients of a complex power series are its derivatives at the centre divided by the corresponding factorials
Statement
If near , then for every ,
Facts & Assumptions
Given: A complex power-series representation of about .
The th derivative is obtained by repeated termwise differentiation with falling-factorial coefficients (A complex power-series sum has complex derivatives of every order, obtained by repeated termwise differentiation).
Complex natural powers are defined by the recursion and for ; in particular (Integer powers in the complex field).
The factorial is nonzero (The factorial and the falling factorial , defined by recursion in ).
Proof
Evaluate [L1] at , so every remaining power is with . From the recursion of [L2], for every , so for every positive , while by the base clause of [L2]. Hence every term with a positive remaining power vanishes and the term indexed by is .
Thus ; divide by the nonzero factorial from [L3]. For , this reads .
Depends on
Used by
- A complex power-series representation about a fixed centre has unique coefficients Corollary
- A flat smooth real function has no holomorphic extension near zero Counterexample
- sin z-z has a zero of order three at the origin Example
- Agreement of the power-series and Cauchy-integral formulas for Taylor coefficients Remark
- A complex power-series sum re-expands about every interior point, at least to the distance from that point to the original boundary Theorem
- A power series of finite radius has a singular point on its circle of convergence Theorem
- The order of a zero is the exponent in its local holomorphic factorization Theorem
Dependency tree · two levels
22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- L. Ahlfors, Complex Analysis, 3rd ed., Ch. 2 (standard reference, not scraped)