Complex Analysis

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

Class 31 (Friday April 10)

SUMMARY Applications of Residues to Real Integrals CURRENT READING Brown & Curchill pages 210�219 NEXT READING Brown & Curchill 204�210

The beauty of Complex Residue Calculus is that it allows us to evaluate a vast number of contour integrals. In fact, we can show that we can use residues to evaluate associated real integrals which would otherwise be very dif�cult to get exact values for become quite easy as contour integrals.

Recall the de�nition of z = e^{iθ} = cos(θ) + i sin(θ) Therefore, we can write cos(θ) and sin(θ) in terms of z. cos θ = sin θ =

\underline{Exercise}

Rewrite the integral ∫_{0}^{2π} \frac{dθ}{3 + 2 sin θ} in terms of z, using z = e^{iθ} where 0 ≤ θ ≤ 2π.

Evaluate the real integral by evaluating the contour integral.

\underline{Groupwork}

Show that ∫_{0}^{2π} \frac{cos 3θ}{5 − 4 cos θ}dθ = \frac{i}{2} ∮_{|z|=1} \frac{z^{6} + 1}{z^{3}(2z^{2} − 5z + 2)}dz

You should be able to compute the value of the contour integral and show it equals \frac{π}{12}.

Evaluation of real integrals of the form ∫_{−∞}^{∞} f(x) dx

We can also evaluate Improper Integrals more easily by evaluating associated contour integrals. however, we have to have some conditions on the integrand f(z). Boundedness Theorem 1 If |f(z)| ≤ \frac{M}{R^{k}} for z = Re^{iθ} where k > 1 and M are constants and C_{R} is the contour

consisting of the semi-circle of radius R from θ = 0 to θ = π, then lim_{R→∞} ∫_{C_{R}} f(z)e^{inz} dz = 0

Boundedness Theorem 2 If |f(z)| ≤ \frac{M}{R^{k}} for z = Re^{iθ} where k > 1, n > 0 and M are constants and C_{R} is the contour

consisting of the semi-circle of radius R from θ = 0 to θ = π, then lim_{R→∞} ∫_{C_{R}} f(z)e^{inz} dz = 0

The second boundedness theorem is sometimes called Jordan’s Lemma.

You can use these theorems above to say that

∫_{−R}^{R} f(x)dx + ∫_{C_{R}} f(z) dz = ∮_{C} f(z) dz

and then take the limit as R → ∞ to say that

∫_{−∞}^{∞} f(x) dx = ∮ f(z) dz = 2πi ∑ Res(f)

if the boundedness theorems above apply to f(z) (since then ∫_{C_{R}} f(z) dz → 0 as R → ∞) \underline{Examples}

Show that ∫_{−∞}^{∞} \frac{1}{(x^{2} + 1)^{2}} dx = \frac{π}{2}

Show that (for a > 0) ∫_{0}^{∞} \frac{cos(ax)}{x^{2} + 1} dx = \frac{π}{2}e^{−a}