# Logarithmic residue

of a meromorphic function $w = f ( z)$ at a point $a$ of the extended complex $z$- plane

The residue

$$\mathop{\rm res} _ {a} \ \frac{f ^ { \prime } ( z) }{f ( z) }$$

of the logarithmic derivative $f ^ { \prime } ( z) / f ( z)$ at the point $a$. Representing the function $\mathop{\rm ln} f ( z)$ in a neighbourhood $V ( a)$ of a point $a \neq \infty$ in the form $\mathop{\rm ln} f ( z) = A \mathop{\rm ln} ( z - a ) + \phi ( z)$, where $\phi ( z)$ is a regular function in $V ( a)$, one obtains

$$\mathop{\rm res} _ {a} \ \frac{f ^ { \prime } ( z) }{f ( z) } = A .$$

The corresponding formulas for the case $a = \infty$ have the form

$$\mathop{\rm ln} f ( z) = A \mathop{\rm ln} \left ( \frac{1}{z} \right ) + \phi ( z) ,$$

$$\mathop{\rm res} _ \infty \frac{f ^ { \prime } ( z) }{f ( z) } = A .$$

If $a$ is a zero or a pole of $f ( z)$ of multiplicity $m$, then the logarithmic residue of $f ( z)$ at $a$ is equal to $m$ or $- m$, respectively; at all other points the logarithmic residue is zero.

If $f ( z)$ is a meromorphic function in a domain $D$ and $\Gamma$ is a rectifiable Jordan curve situated in $D$ and not passing through the zeros or poles of $f ( z)$, then the logarithmic residue of $f ( z)$ with respect to the contour $\Gamma$ is the integral

$$\tag{1 } \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{f ^ { \prime } ( z) }{f ( z) } \ d z = N - P ,$$

where $N$ is the number of zeros and $P$ is the number of poles of $f ( z)$ inside $\Gamma$( taking account of multiplicity). The geometrical meaning of (1) is that as $\Gamma$ is traversed in the positive sense, the vector $w = f ( z)$ performs $N - P$ rotations about the origin $w = 0$ of the $w$- plane (see Argument, principle of the). In particular, if $f ( z)$ is regular in $D$, that is, $P = 0$, then from (1) one obtains a formula for the calculation of the index of the point $w = 0$ with respect to the image $\Gamma ^ {*} = f ( \Gamma )$ of $\Gamma$ by means of the logarithmic residue:

$$\tag{2 } \mathop{\rm ind} _ {0} \Gamma ^ {*} = \ \frac{1}{2 \pi i } \int\limits _ \Gamma \frac{f ^ { \prime } ( z) }{f ( z) } \ d z .$$

Formula (2) leads to a generalization of the concept of a logarithmic residue to regular functions of several complex variables in a domain $D$ of the complex space $\mathbf C ^ {n}$, $n \geq 1$. Let $w = f ( z) = ( f _ {1} \dots f _ {n} ) : D \rightarrow \mathbf C ^ {n}$ be a holomorphic mapping such that the Jacobian $J _ {f} ( z) \not\equiv 0$ and the set of zeros $E = f ^ { - 1 } ( 0)$ is isolated in $D$. Then for any domain $G \subset \overline{G}\; \subset D$ bounded by a simple closed surface $\Gamma$ not passing through the zeros of $f$ one has a formula for the index of the point $w = 0$ with respect to the image $\Gamma ^ {*} = f ( \Gamma )$:

$$\tag{3 } \mathop{\rm ind} _ {0} \Gamma ^ {*} = \ \frac{1}{( 2 \pi i ) ^ {n} } \int\limits _ {\Gamma _ \epsilon } \frac{d f _ {1} \wedge \dots \wedge d f _ {n} }{f _ {1} \dots f _ {n} } = N ,$$

where the integration is carried out with respect to the $n$- dimensional frame $\Gamma _ \epsilon = \{ {z \in G } : {| f _ \nu ( z) | = \epsilon, \nu = 1 \dots n } \}$ with sufficiently small $\epsilon > 0$. The integral in (3) also expresses the sum of the multiplicities of the zeros of $f$ in $G$( see ).

How to Cite This Entry:
Logarithmic residue. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logarithmic_residue&oldid=47703
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article