# variant of Cauchy integral formula

Theorem. Let $f(z)$ be holomorphic in a domain $A$ of $\u2102$. If $C$ is a closed contour not intersecting itself which with its domain is contained in $A$ and if $z$ is an arbitrary point inside $C$, then

$f(z)={\displaystyle \frac{1}{2i\pi}}{\displaystyle {\oint}_{C}}{\displaystyle \frac{f(t)}{t-z}}\mathit{d}t.$ | (1) |

*Proof.* Let $\epsilon $ be any positive number. Denote by ${C}_{r}$ the circles with radius $r$ and centered in $z$. We have

$${\oint}_{C}\frac{f(t)}{t-z}\mathit{d}t={\oint}_{C}\frac{f(z)+(f(t)-f(z))}{t-z}\mathit{d}t=\underset{I}{\underset{\u23df}{{\oint}_{C}\frac{f(z)}{t-z}\mathit{d}t}}+\underset{J}{\underset{\u23df}{{\oint}_{C}\frac{f(t)-f(z)}{t-z}\mathit{d}t}}.$$ |

According to the corollary of Cauchy integral theorem and its example, we may write

$$I=f(z){\oint}_{C}\frac{dt}{t-z}=\mathrm{\hspace{0.33em}2}i\pi f(z).$$ |

If $$, we have

$$J={\oint}_{{C}_{r}}\frac{f(t)-f(z)}{t-z}\mathit{d}t.$$ |

The continuity of $f$ in the point $z$ implies, that

$$ |

when $$ i.e. when

$$ | (2) |

If (2) is in , we obtain first

$$ |

whence, by the estimation theorem of integral,

$$ |

and lastly

$$ | (3) |

This result implies (1).

Title | variant of Cauchy integral formula |

Canonical name | VariantOfCauchyIntegralFormula |

Date of creation | 2013-03-22 18:54:15 |

Last modified on | 2013-03-22 18:54:15 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 30E20 |

Synonym | Cauchy integral formula^{} |

Related topic | CauchyIntegralFormula |

Related topic | CorollaryOfCauchyIntegralTheorem |

Related topic | ExampleOfFindingTheGeneratingFunction |

Related topic | GeneratingFunctionOfLaguerrePolynomials |

Related topic | GeneratingFunctionOfHermitePolynomials |