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Agreement of the power-series and Cauchy-integral formulas for Taylor coefficients
Remark
For the Taylor series of The Taylor series of a holomorphic function at a point, the coefficient of is . This agrees with the coefficient formula for an arbitrary convergent complex power-series representation in The coefficients of a complex power series are its derivatives at the centre divided by the corresponding factorials. If is small enough that the circle and its interior lie in the holomorphy domain, All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle gives the same coefficient as
Thus the derivative and contour formulas name the coefficients of the expansion established by A holomorphic function equals its Taylor series throughout the largest centred disc in its domain; neither is an additional choice of series.
Depends on
- The Taylor series of a holomorphic function at a point
- A holomorphic function equals its Taylor series throughout the largest centred disc in its domain
- The coefficients of a complex power series are its derivatives at the centre divided by the corresponding factorials
- All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle
Used by
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Dependency tree · two levels
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Sources
- Lars Ahlfors, Complex Analysis, 3rd ed., Ch. 5 §1.2 (standard reference, not scraped)
- Matthias Weber, Complex Analysis, §2.2 (standard reference, not scraped)
- B. V. Shabat, Introduction to Complex Analysis, Ch. 2 (standard reference, not scraped)