$g:\mathbb{C}\rightarrow\mathbb{C}$ be analytic and $g(0)\neq 0$ we need to calculate $\frac{1}{2\pi i}\int_{|z|=r>0} f(z)g(z) dz$

Question

$f:\mathbb{C}^{*}\to\mathbb{C}$ be a analytic with a simple pole of order $1$ at $0$ wit residue $a_{-1}$.

$g:\mathbb{C}\rightarrow\mathbb{C}$ be analytic and $g(0)\neq 0$ we need to calculate $$\frac{1}{2\pi i}\int_{|z|=r>0} f(z)g(z) dz$$

Please give me Hints.

Calculate residue at 0 using Laurent series? – akkkk Mar 19 '13 at 11:25 — akkkk, Mar 19 '13 at 11:25

score 1 · Accepted Answer · answered Mar 19 '13 at 11:41

In a some neighbourhood $U_r(0)$ function $f$ can be represented as $f(z)=\dfrac{\varphi(z)}{z},$ where $\varphi(z)$ is holomorphic in $U_r(0)$ and $\varphi(z)\ne{0}.$ At conditions on $f$ $$\varphi(z)=a_{-1}+a_0 z+a_1z^2+\ldots$$ therefore $$\frac{1}{2\pi i}\int\limits_{|z|=r>0} f(z)g(z) dz=\frac{a_{-1}}{2\pi i}\int\limits_{|z|=r>0} \frac{g(z)}{z} dz=\ldots$$

$g:\mathbb{C}\rightarrow\mathbb{C}$ be analytic and $g(0)\neq 0$ we need to calculate $\frac{1}{2\pi i}\int_{|z|=r>0} f(z)g(z) dz$

1 Answers1