In complex analysis, a complexvalued function f of a complex variable z
One of the most important theorems of complex analysis is that holomorphic functions are analytic. Among the corollaries of this theorem are
The argument, first given by Cauchy, hinges on Cauchy's integral formula and the power series development of the expression
Suppose ƒ is differentiable everywhere within some open disk centered at a. Let z be within that open disk. Let C be a positively oriented (i.e., counterclockwise) circle centered at a, lying within that open disk but farther from a than z is. Starting with Cauchy's integral formula, we have
The interchange of the sum and the integral is justified by the uniform convergence of the geometric series within subsets of its disk of convergence that are bounded away from the boundary. Since the factor (z − a)^{n} does not depend on the variable of integration w, it can be pulled out:
And now the integral and the factor of 1/(2πi) do not depend on z, i.e., as a function of z, that whole expression is a constant c_{n}, so we can write:
and that is the desired power series.
In complex analysis, a field of mathematics, a complexvalued function f of a complex variable z
One of the most important theorems of complex analysis is that holomorphic functions are analytic. Among the corollaries of this theorem are
The argument, first given by Cauchy, hinges on Cauchy's integral formula and the power series development of the expression
Suppose f is differentiable everywhere within some open disk centered at a. Let z be within that open disk. Let C be a positively oriented (i.e., counterclockwise) circle centered at a, lying within that open disk but farther from a than z is. Starting with Cauchy's integral formula, we have
&{}= {1 \over 2\pi i}\int_C {1 \over wa}\cdot{wa \over wz}f(w)\,dw \\ \\ &{}= {1 \over 2\pi i}\int_C {1 \over wa}\cdot{wa \over (wa)(za)}f(w)\,dw \\ \\ &{}={1 \over 2\pi i}\int_C {1 \over wa}\cdot{1 \over 1{za \over wa}}f(w)\,dw \\ \\ &{}={1 \over 2\pi i}\int_C {1 \over wa}\cdot{\sum_{n=0}^\infty\left({za \over wa}\right)^n} f(w)\,dw \\ \\ &{}=\sum_{n=0}^\infty{1 \over 2\pi i}\int_C {(za)^n \over (wa)^{n+1}} f(w)\,dw.\end{align}
The interchange of the sum and the integral is justified by the uniform convergence of the geometric series within subsets of its disk of convergence that are bounded away from the boundary. Since the factor (z − a)^{n} does not depend on the variable of integration w, it can be pulled out:
And now the integral and the factor of 1/(2πi) do not depend on z, i.e., as a function of z, that whole expression is a constant c_{n}, so we can write:
and that is the desired power series.
