Complex Analysis

Math 312  MWF 10:30am - 11:25am
\begin{matrix}c \\ ◯ \end{matrix}1998 Ron Buckmire  Fowler 112

Class 28 (Friday April 3)

SUMMARY Poles, Zeroes, Residues and Cauchy’s Residue Theorem CURRENT READING Brown & Curchill pages 193-196 NEXT READING Brown & Curchill pages 180-196

Zeroes and Poles So, we have had a lot of experience �nding the poles of a function. The problem of �nding a pole is equivalent to �nding the zero of a related function. Let’s formalize these de�nitions: Zero A point z_{0} is called a zero of order m for the function f(z) if f is analytic at z_{0} and f and its �rst m − 1 derivatives vanish at z_{0}, but f^{(m)}(z_{0}) ≠ 0. Pole A point z_{0} is called a pole of order m of f(z) if 1/f has a zero of order m at z_{0}. Identifying Poles and Zeroes Let f be analytic. Then f has a zero of order mat z_{0} if and only if f(z) can be written as f(z) = g(z)(z − z_{0})^{m} where g is analytic at z_{0} and g(z_{0}) ≠ 0.

If f(z) can be written as f(z) = \frac{g(z)}{(z − z_{0})^{m}} where g(z) is analytic at z_{0}, then f has a pole of

order m at z = z_{0} and g(z_{0}) ≠ 0

How do we �nd poles of a function? Well, if we have a quotient function f(z) = p(z)/q(z) where p(z) and q(z) are analytic at z_{0} and p(z_{0}) ≠ 0 then f(z) has a pole of order m if and only if q(z) has a zero of order m. \underline{Exercise}

We will classify all the singularities of f(z) = \frac{3z + 2}{z^{2} + 1}

\underline{Groupwork} Let’s try and classify all the singularities of the following functions:

(a) A(z) = \frac{4}{z^{2}(z − 1)^{3}}

(b) B(z) = \frac{sin z}{z^{2} − 4}

(c) C(z) = tanz

(d) D(z) = \frac{z}{z^{2} − 6z + 10}

\underline{Residues} Once we know all the singularities of a function it is useful to compute the residues of that function. If a function f(z) has a pole of order m at z_{0}, the residue, denoted by Res(f; z_{0}) or Res(z_{0}) is given by the formula below:

Res(f; z_{0}) = lim_{z→z_{0}}\{\frac{1}{(m − 1)!} \frac{d^{m−1}}{dz^{m−1}}[(z − z_{0})^{m}f(z)]\}

Find the residues of all the singularities we previously classi�ed. (a)

(b)

(c)

(d)

Cauchy’s Residue Theorem If f is analytic on a simple (positively oriented) closed contour and everywhere inside except the �nite number of points z_{1}, z_{2}, ⋯ z_{n} inside , then

∮ f(z) dz = 2πi ∑_{k=1}^{n}Res(f; z_{k})

Use Cauchy Residue Theorem to evaluate the following integrals:

(a) ∮_{|z|=2} \frac{3z + 2}{z^{2} + 1}

(b) ∮_{|z|=5} \frac{4}{z^{2}(z − 1)^{3}} dz

(c) ∮_{|z|=5π} \frac{sin z}{z^{2} − 4} dz

(d) ∮_{|z|=2π} tanz dz
(e) ∮_{|z|=8} \frac{z}{z^{2} − 6z + 10} dz