Definition: Residue. Consider the function \(f(z)\) with an isolated singularity at \(z_0\), i.e. defined on the region \(0 < |z - z_0| < r\) and with Laurent series (on that region) \[f(z) = \sum_{n = 1}^{\infty} \dfrac{b_n}{(z - z_0)^n} + \sum_{n = 0}^{\infty} a_n (z - z_0)^n. \nonumber \] The residue of \(f\) at \(z_0\) is \(b_1\). This is ...

Find the residue of $f(z)=\frac{e^{az}}{1+e^z}$. at $\pi i$ 1 Finding residue of complex function, the result is different when using Laurent series and residue theorem.

Behavior of functions near zeros and poles. The basic idea is that near a zero of order , a function behaves like ( − 0) and near a pole of order , a function behaves like 1∕( − 0) . The following make this a little more precise. Behavior near a zero. If has a zero of order at 0. then near 0, ( ) ø ( − 0) ,

residue first obtains the poles using roots. Next, if the fraction is nonproper, the direct term k is found using deconv, which performs polynomial long division. Finally, residue determines the residues by evaluating the polynomial with individual roots removed.

In this section, we will discuss graphical methods for obtaining the residue i.e. partial-fraction expansion coefficients directly from the pole-zero plot. Consider the network function as follows, Then we can expand F (s) as, The poles and zeros are plotted on the complex plane as shown in the Fig. 3.28.