MATH3401

Lecture 36

Applications contour integrals

Example 2 (cont): From Lecture 35, we need limit as $$\rho \downarrow 0$$ and $$R\ra \infty$$.

$I_1 + I_3 \ra \ds 2 i \int_{0}^{\infty} \frac{\sin x}{x}dx .$

Thus $\int_0^{\infty} \frac{\sin x}{x}dx = \frac{-1}{2i} \left( \lim_{\rho \downarrow 0} I_2 + \lim_{R\ra \infty} I_4 \right) \quad (\text{😀})$

if these limits exist.

Applications contour integrals

Example 2 (cont):

$$I_2 = \ds \int_{C_{\rho}}\frac{e^{iz}dz}{z}$$ $$= \ds \int_{\pi}^{0}\frac{e^{i\rho e^{i\theta}}}{\rho e^{i\theta}} i\rho e^{i\theta}d\theta$$, $$\;(z= \rho e^{i\theta}$$ & $$\;\theta: \pi \ra 0)$$

$$= - i \ds \int_{0}^{\pi} e^{i \rho e^{i\theta}}d\theta. \qquad \qquad$$

Since $$\left|\rho e^{i \theta}\right|=\rho$$, $$e^{i \rho e^{i\theta}}\ra 1$$ as $$\rho \ra 0$$ uniformly in $$\theta$$ for $$\theta\in [0,2\pi]$$.

Applications contour integrals

Example 2 (cont): Then $$\ds \lim_{\rho \downarrow 0} I_2 = -i \int_0^{\pi} \lim_{\rho \downarrow 0} \left(e^{i\rho e^{i\theta}}\right)d\theta.$$

Thus $$I_2 \ra -i\ds \int_0^{\pi} 1 d\theta = -i \pi$$ as $$\rho \downarrow 0$$.

Also, $$I_4 \ra 0$$ as $$R \ra \infty$$ by Jordan's Lemma, with $$\alpha=\beta=M=1$$.

So $$(\text{😀})$$ implies

$$\ds \int_0^{\infty} \frac{\sin x}{x} dx$$ $$= \ds-\frac{1}{2i}\big[-i\pi +0\big]$$ $$= \ds\frac{\pi}{2}$$.

Jordan's Lemma

Lemma: If $$f$$ is analytic on $$\left\{ z : \Im z \ge 0\right\} \cap \left\{ z : |z| \gt R_0 \right\}$$ satisfies $$\big|\,f(z) \le \frac{M}{R^\beta}\big|\;$$ $$(*)\;$$ ($$M,\beta \gt 0$$ constants) on $\Gamma_R=\left\{ Re^{i\theta}: R\gt R_0, 0\le \theta \le \pi \right\}.$ Then $\ds \lim_{R\ra \infty } \int_{\Gamma_R } e^{i\alpha z} f(z)\, dz = 0,\; \forall \alpha\gt 0.$

Argument principle

Consider $$f$$ analytic in and on $$C$$, a simple closed curve except possibly for poles inside $$C$$.

Argument principle: $$\Delta_{C} \arg f = 2\pi (z-p)$$

$$\quad z$$: # of zeros inside $$C$$, counting multiplicity

$$\quad p$$: # of poles inside $$C$$, counting total order

E. g. poles of order 3 & 5 inside $$C \Rightarrow p=8$$.

Rouche's Theorem

Theorem: Let $$f$$ and $$g$$ be analytic in and on a simple closed curve $$C$$, orientation irrelevant. Suppose $$|\,f(z)|\gt |g(z)|,\; \forall z\in \C$$. Then $$f$$ and $$f+g$$ have the same number of zeros (counting multiplicity) inside $$C$$.

Nomeclature: $$f(z)=(z-i)^2(z+i)^3$$ has 5 zeros (counting multiplicity).

Rouché's Theorem

Application: How many zeros of $$h(z) = z^7 -4z^3 +z-1$$ lie inside the unit circle?

Put $$\;f = -4z^3 \text{ and } g = z^7+z-1$$. Notice that $$f$$ and $$g$$ are entire (polynomials).

$$|\,f|=4$$ on $$C$$ and $$|g(z)|\le |z^7| +|z| +|-1|$$ $$=3$$ $$\lt 4 = |\, f(z)|$$.

So $$|f| \lt |g|$$ on $$C$$. So Rouché's Theorem implies $$f$$ and $$f+g = h$$ have the same number of zeros inside $$C$$, namely 3 ($$f$$ has a 3-fold zero at $$0$$).