Analytic Function

Analytic series are defined  as per the converging series; that revolves around a particular x for which the series has been expanded. It is defined as infinitely differential function, over a variable called x the expanded Taylor series is an example of this can be given below.

\(T(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!}(x-x_{0})^{n}\)

This explains the expanded Taylor  over value Xo; this function is said to be an analytic function for the value x in its domain there is another value in domain which is  that converges the series to one point 

Complex Analytic Function

A function is said to be analytic in the region T of complex plane x if, f(x) hasl derivative at each and every point of x and f(x) has unique values that are it follows one to one function.’

This example explains the analytic function on the complex plane. 

Let \(f:C\rightarrow C\) be an analytic function. For \(z = x + iy\), let \(u, v: R^{2}\) be such that \(u(x, y) = Re f(z)\) and \(v(x, y) = lm\; f(z)\). Which of the following are correct?

\(\frac{\partial ^{2}u}{\partial ^{2}x}+\frac{\partial ^{2}u}{\partial ^{2}y}=0\)

\(\frac{\partial ^{2}v}{\partial ^{2}x}+\frac{\partial ^{2}v}{\partial ^{2}y}=0\)

\(\frac{\partial ^{2}u}{\partial x \partial y}+\frac{\partial ^{2}u}{\partial y \partial x}=0\)

\(\frac{\partial ^{2}v}{\partial x \partial y}+\frac{\partial ^{2}v}{\partial y \partial x}=0\)

Real Analytic Function

If a series agrees with Taylor series and has derivative of different order on its every domain point then that series  is said to be real analytic function.

\(T_{f} = \sum_{\infty }^{k=0}\frac{(z-c)^{k}}{2\pi i}\int _{\gamma }\frac{f(w)}{(w-c)^{k+1}}dw\)

\(= \frac{1}{2\pi i}\int _{\gamma }\frac{f(w)}{w-c}\sum_{k=0}^{\infty}\left ( \frac{z-c}{w-c} \right )^{k}dw\)

\(= \frac{1}{2\pi i}\int _{\gamma }\frac{f(w)}{w-c}\left ( \frac{1}{1-\frac{z-c}{w-c}} \right )dw\)

\(= \frac{1}{2\pi i}\int _{\gamma }\frac{f(w)}{w-z}dw=f(z)\)<

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