# complex logarithm

The $z$ is defined as every complex number^{} $w$ which satisfies the equation

${e}^{w}=z.$ | (1) |

This is is denoted by

$$\mathrm{log}z:=w.$$ |

The solution of (1) is obtained by using the form ${e}^{w}=r{e}^{i\phi}$ , where $r=|z|$ and $\phi =\mathrm{arg}z$; the result is

$$w=\mathrm{log}z=\mathrm{ln}|z|+i\mathrm{arg}z.$$ |

Here, the $\mathrm{ln}|z|$ means the usual Napierian or natural logarithm^{} (http://planetmath.org/NaturalLogarithm2) (‘logarithmus naturalis’) of the real number $|z|$. If we fix the phase angle $\phi $ of $|z|$ so that $$, we can write

$$\mathrm{log}z=\mathrm{ln}r+i\phi +n\cdot 2\pi i\mathit{\hspace{1em}}(n=0,\pm 1,\pm 2,\mathrm{\dots}).$$ |

The complex logarithm $\mathrm{log}z$ is defined for all $z\ne 0$ and it is infinitely multivalued $-$ e.g. $\mathrm{log}(-1)=(2n+1)\pi i$ where $n$ is an arbitrary integer. The values with $n=0$ are called the of the ; if $z$ is real, the value of $\mathrm{log}z$ coincides with $\mathrm{ln}z$.

Title | complex logarithm |

Canonical name | ComplexLogarithm |

Date of creation | 2013-03-22 14:43:11 |

Last modified on | 2013-03-22 14:43:11 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 10 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 32A05 |

Classification | msc 30D20 |

Synonym | natural logarithm |

Related topic | Logarithm |

Related topic | NaturalLogarithm2 |

Related topic | ValuesOfComplexCosine |

Related topic | EqualityOfComplexNumbers |

Related topic | SomeValuesCharacterisingI |

Related topic | UsingResidueTheoremNearBranchPoint |