# second form of Cauchy integral theorem

###### Theorem.

Let the complex function $f$ be analytic in a simply connected open domain $U$ of the complex plane^{}, and let $a$ and $b$ be any two points of $U$. Then the contour integral

${\int}_{\gamma}}f(z)\mathit{d}z$ | (1) |

is independent on the path $\gamma $ which in $U$ goes from $a$ to $b$.

Example. Let’s consider the integral (1) of the real part function defined by

$$f(z):=\text{Re}(z)$$ |

with the path $\gamma $ going from the point $O=(0,\mathrm{\hspace{0.17em}0})$ to the point $Q=(1,\mathrm{\hspace{0.17em}1})$. If $\gamma $ is the line segment^{} $OQ$, we may use the substitution

$$z:=(1+i)t,dz=(1+i)dt,0\leqq t\leqq 1,$$ |

and (1) equals

$${\int}_{0}^{1}t\cdot (1+i)\mathit{d}t=\frac{1}{2}+\frac{1}{2}i.$$ |

Secondly, we choose for $\gamma $ the broken line $OPQ$ where $P=(1,\mathrm{\hspace{0.17em}0})$. Now (1) is the sum

$${\int}_{OP}\text{Re}(z)\mathit{d}z+{\int}_{PQ}\text{Re}(z)\mathit{d}z={\int}_{0}^{1}x\mathit{d}x+{\int}_{0}^{1}i\mathit{d}y=\frac{1}{2}+i.$$ |

Thus, the integral (1) of the function depends on the path between the two points. This is explained by the fact that the function $f$ is not analytic — its real part $x$ and imaginary part 0 do not satisfy the Cauchy-Riemann equations^{}.

Title | second form of Cauchy integral theorem |
---|---|

Canonical name | SecondFormOfCauchyIntegralTheorem |

Date of creation | 2013-03-22 15:19:39 |

Last modified on | 2013-03-22 15:19:39 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 30E20 |

Synonym | equivalent form of Cauchy integral theorem |

Related topic | CauchyIntegralTheorem |

Defines | example of non-analytic function |