# MATH3401

Lecture 34

### Residues at Poles

Example 1:   $$f(z)=\dfrac{z+i}{z^2+9}$$.  $$f$$ is analytic on $$\C\setminus\{\pm 3i\}$$.

$$\pm 3i$$ are isolated singularities.

Near $$z=3i$$, write $f(z) = \frac{\phi(z)}{z-3i}$ where $$\phi(z) = \dfrac{z+i}{z+3i}$$ is analytic and non-zero near $$z=3i$$.

### Residues at Poles

Example 1 (cont): Theorem 2 (Lec 33) $$\Rightarrow$$ simple pole at $$z=3i$$, and $$(**) \Rightarrow$$

$$\underset{z=3i}{\res} \,f (z) =\phi(3i)$$ $$= \dfrac{2}{3}$$.

📝 Do for $$z=-3i$$.

### Residues at Poles

Example 2:   $$f(z)=\dfrac{z^3+2z}{(z-i)^3}$$.  $$f$$ is analytic on $$\C\setminus\{ i \}$$.

Near $$z=i$$, we have $$f(z) = \dfrac{\phi(z)}{(z-i)^3}$$ where $$\phi(z) = z^3+2z$$ is analytic and $$\phi(i)= i \neq 0$$.

Theorem 1 (Lec 33) $$\Rightarrow$$ pole of order 3 at $$z=i$$, so

$$\underset{z=i}{\res} \,f(z)=\dfrac{\phi^{(2)}(i)}{2!}$$ $$= 3i$$.

### Zeros of analytic functions

Suppose that $$f$$ analytic at $$z_0$$:   $$f$$ has a zero of order $$m$$ at $$z_0$$ if $\left\{ \begin{array}{lr} f^{\left(j\right)}(z_0) = 0 & j=0, 1, \ldots , m-1\\ f^{\left(m\right)}(z_0) \neq 0 & \end{array} \right.$

E. g. $$f(z)=(z-i)^4(z-4)$$ has a zero of order $$4$$ at $$i$$ and a simple zero at $$4$$.

### Zeros of analytic functions

Theorem 1: $$f$$ analytic at $$z_0$$ has a zero of order $$m$$ at $$z_0$$ if and only if $f(z)=(z-z_0)^mg(z)$ with $$g$$ analytic and $$g(z_0)\neq 0$$.

### Zeros and Poles

Theorem 2: Suppose $$p$$ and $$q$$ are analytic at $$z_0$$, $$p(z_0)\ne 0,$$ $$q$$ has a zero of order $$m$$ at $$z_0$$. Then $$p/q$$ has a pole of order $$m$$ at $$z_0$$.

### Zeros and Poles

Example: $$p(z)=1$$ and $$q(z)=z(e^z-1)$$. $$p/q$$ has an isolated singularity at $$0$$. $$p$$ is analytic and nonzero.

$$q(0)=0, \;\quad\quad$$

$$q'(0)= 0, \quad\;\;$$ 📝

$$q''(0)= 2\neq 0$$. 📝

Theorem 2 $$\Rightarrow$$ has a pole of order $$2$$ at $$0$$.

### Zeros and Poles

Theorem 3: Let $$p$$, $$q$$ be analytic at $$z_0$$. If $$p(z_0)\neq 0$$, $$q(z_0)=0$$, $$q'(z_0)\neq 0$$. Then $$p/q$$ has a simple at $$z_0$$, and $\underset{z=z_0}{\res}\frac{p(z)}{q(z)} = \frac{p(z_0)}{q'(z_0)}$

Note: There exist higher order analogues but they are messy.

### Applications contour integrals

Recall: ${\small\int_{-\infty}^{\infty} f(x)\,dx = \lim_{M_1\ra -\infty}\int_{M_1}^{0} f(x)\,dx + \lim_{M_2\ra \infty}\int_{0}^{M_2} f(x)\,dx}$ if both of the limits exist.

Remark: We can replace $$0$$ by any fixed $$c\in \R$$.

### Applications contour integrals

You cannot in general replace RHS by $\lim_{M\ra -\infty}\int_{-M}^{M} f(x)\,dx .$

If you do this anyhow, it defines the Cauchy Principal value (PV) integral.

### Applications contour integrals

E. g. $$\ds\int_{-\infty}^{\infty}x \,dx$$ $$=\ds\lim_{M_1\ra -\infty}\int_{M_1}^{0} x\,dx + \lim_{M_2\ra \infty}\int_{0}^{M_2} x\,dx$$

$$\qquad\qquad\quad = \ds\lim_{M_1\ra -\infty} \frac{-M_1^2}{2} + \lim_{M_2\ra \infty} \frac{M_2^2}{2}$$

$$\qquad\qquad\quad =$$ undefined, but

$$\ds\text{PV} \int_{-\infty}^{\infty}x\, dx = \lim_{M\ra \infty}\int_{-M}^{M} x\,dx$$ $$=\ds \lim_{M\ra \infty}\left[\frac{M^2}{2}-\frac{M^2}{2}\right]$$ $$=\ds \lim_{M\ra \infty}\left[0\right]$$ $$=0$$.

### Applications contour integrals

When is $$\text{PV}\ds \int_{-\infty}^{\infty} f = \int_{-\infty}^{\infty}f\,$$? E. g. for $$f$$ even (or for $$f\geq 0$$).

If $$f$$ is even, i. e. $$f(x) = f(-x)$$ $$\forall x\in \R$$, then

$$\ds\int_{0}^{\infty} f(x)\, dx = \frac{1}{2} \int_{-\infty}^{\infty} f(x)\, dx$$ $$=\ds \frac{1}{2}\text{PV} \int_{-\infty}^{\infty} f(x)\, dx$$

and all of these converge or diverge together.