﻿ 柯西积分公式的一点注记 A Note on Cauchy Integral Formula

Pure Mathematics
Vol. 09  No. 03 ( 2019 ), Article ID: 30088 , 5 pages
10.12677/PM.2019.93037

A Note on Cauchy Integral Formula

Hongying Si

College of Math Mathematics and Statistics, Shangqiu Normal University, Shangqiu Henan

Received: Apr. 14th, 2019; accepted: Apr. 25th, 2019; published: May 6th, 2019

ABSTRACT

In this paper, according to integral calculation based on ${\int }_{c}\frac{1}{{\left(z-{z}_{0}\right)}^{n}}dz$ in Example 3.2, the integral value of Example 3.2 is calculated by the parametric equation method and the case 3.2 is generalized from the integral curve and the integrand function. First, the integral curve is generalized, and the circle with ${z}_{0}$ as the center and r as the radius is generalized to any closed curve containing ${z}_{0}$ ; after the promotion, this example has a wider scope of application. Secondly, the integrand function is promoted, $\frac{1}{z-{z}_{0}}$ is promoted to $\frac{f\left(z\right)}{z-{z}_{0}}$ and $\frac{f\left(z\right)}{{\left(z-{z}_{0}\right)}^{n+1}}$ respectively, the close relationship between the case 3.2 and the Cauchy integral formula and the high-order derivative formula of the analytic function is discussed further.

Keywords:Integral Curve, Cauchy Integral Formula, Higher Derivative Formula

1. 引言

2. 回顾例3.2

${\int }_{c}f\left(z\right)\text{d}z={\int }_{c}u\text{d}x-v\text{d}y+i{\int }_{c}v\text{d}x+u\text{d}y.$ (1)

$\begin{array}{c}{\int }_{c}f\left(z\right)\text{d}z={\int }_{c}\left(u+iv\right)\text{d}\left(x+iy\right)\\ ={\int }_{c}\left(u+iv\right)\left(\text{d}x+i\text{d}y\right)\\ ={\int }_{c}u\text{d}x+iu\text{d}y+iv\text{d}x-v\text{d}y\\ ={\int }_{c}u\text{d}x-v\text{d}y+i{\int }_{c}v\text{d}x+u\text{d}y\end{array}$

$\begin{array}{c}{\int }_{c}f\left(z\right)\text{d}z={\int }_{c}u\text{d}x-v\text{d}y+i{\int }_{c}v\text{d}x+u\text{d}y\\ ={\int }_{\alpha }^{\beta }\left[u\left(t\right){x}^{\prime }\left(t\right)-v\left(t\right){y}^{\prime }\left(t\right)\right]\text{d}t+i{\int }_{\alpha }^{\beta }\left[u\left(t\right){y}^{\prime }\left(t\right)+v\left(t\right){x}^{\prime }\left(t\right)\right]\text{d}t\end{array}$

${\int }_{c}f\left(z\right)\text{d}z={\int }_{\alpha }^{\beta }f\left[z\left(t\right)\right]{z}^{\prime }\left(t\right)\text{d}t,$ (2)

${\int }_{c}f\left(z\right)\text{d}z={\int }_{\alpha }^{\beta }\mathrm{Re}\left\{f\left[z\left(t\right)\right]{z}^{\prime }\left(t\right)\right\}\text{d}t+i{\int }_{\alpha }^{\beta }\mathrm{Im}\left\{f\left[z\left(t\right)\right]{z}^{\prime }\left(t\right)\right\}\text{d}t.$ (3)

$\begin{array}{c}{\int }_{c}\frac{1}{{\left(z-{z}_{0}\right)}^{n}}\text{d}z={\int }_{0}^{\text{2π}}\frac{ir{\text{e}}^{i\theta }}{{r}^{n}{\text{e}}^{in\theta }}\text{d}\theta =\frac{i}{{r}^{n-1}}{\int }_{0}^{\text{2π}}{\text{e}}^{-i\left(n-1\right)\theta }\text{d}\theta \\ =\frac{i}{{r}^{n-1}}{\int }_{0}^{\text{2π}}\mathrm{cos}\left(n-1\right)\theta \text{d}\theta +\frac{1}{{r}^{n-1}}{\int }_{0}^{\text{2π}}\mathrm{sin}\left(n-1\right)\theta \text{d}\theta \\ =\left\{\begin{array}{l}2\text{π}i,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n=1;\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\ne 1.\end{array}\end{array}$

3. 例3.2的推广形式

3.1. 将积分曲线推广

${\int }_{c}f\left(z\right)\text{d}z=\underset{k=1}{\overset{n}{\sum }}{\int }_{{c}_{k}}f\left(z\right)\text{d}z$

${\int }_{c}\frac{1}{{\left(z-{z}_{0}\right)}^{n}}\text{d}z={\int }_{{c}_{1}}\frac{1}{{\left(z-{z}_{0}\right)}^{n}}\text{d}z$

${\int }_{c}\frac{1}{{\left(z-{z}_{0}\right)}^{n}}\text{d}z={\int }_{{c}_{1}}\frac{1}{{\left(z-{z}_{0}\right)}^{n}}\text{d}z=\left\{\begin{array}{l}2\text{π}i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n=1,\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\ne 1.\end{array}$

3.2. $n=1$ 时，将被积函数 $\frac{1}{z-{z}_{0}}$ 推广到 $\frac{f\left(z\right)}{z-{z}_{0}}$

$f\left({z}_{0}\right)=\frac{1}{2\text{π}i}{\oint }_{c}\frac{f\left(z\right)}{z-{z}_{0}}\text{d}z$

${\oint }_{c}\frac{f\left(z\right)}{z-{z}_{0}}\text{d}z=2\text{π}if\left(z0\right)$

3.3. $n\ne 1$ 时，将被积函数 $\frac{1}{{\left(z-{z}_{0}\right)}^{n}}$ 推广到 

${f}^{\left(n\right)}\left({z}_{0}\right)=\frac{n!}{2\text{π}i}{\oint }_{c}\frac{f\left(z\right)}{{\left(z-{z}_{0}\right)}^{n+1}}\text{d}z$ $\left(n=1,2,\cdots \right)$

${\oint }_{c}\frac{f\left(z\right)}{{\left(z-{z}_{0}\right)}^{n+1}}\text{d}z=\frac{2\text{π}i}{n!}{f}^{\left(n\right)}\left({z}_{0}\right).$

A Note on Cauchy Integral Formula[J]. 理论数学, 2019, 09(03): 282-286. https://doi.org/10.12677/PM.2019.93037

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