# MATH3401

Lecture 35

### Application: contour integrals

Suppose $$f$$ is even and "nice" on $$\R$$. Our aim is to evaluate $$\ds\int_{-\infty}^{\infty}f$$.

### Applications contour integrals

Suppose $$f$$ is analytic in and on $$C = \Gamma_1+ \Gamma_2$$, except maybe for isolated singularities in $$\text{Int}(C)$$.

$$\ds\int_C f = \int_{\Gamma_1}f + \int_{\Gamma_2}f$$

$$\;\;\qquad(\text{I})$$ $$\qquad (\text{II})$$

LHS: hopefully can evaluate via residue theorem. Then let $$R \ra \infty$$.

### Applications contour integrals

$$(\text{II}) \ra$$ $$\text{PV}\ds\int_{-\infty}^{\infty}f = \int_{-\infty}^{\infty}f$$ since $$f$$ is even.

Only remains to deal with $$\ds\lim_{R\ra \infty}\int_{\Gamma_1}f$$.

Hopefully = 0, e. g. via $$M$$-$$\ell$$ estimate.

### Applications contour integrals

Example 1: Evaluate $$I = \ds \int_{0}^{\infty}\frac{x^2}{x^6+1}dx$$.

Note $$f$$ is even and continuous. Also $$f \backsim 1/x^4$$ as $$x\ra \pm\infty$$.
So $$I$$ converges ($$p$$-test, $$p\gt 1$$).

Note $$f(z)=\dfrac{z^2}{z^6+1}$$ is analytic on $$\C$$, except for the 6 zeros of the denominator $$z^6+1$$.

### Applications contour integrals

Example (cont): $$f$$ is analytic in and on $$C= \Gamma_1 + \Gamma_2$$, except for the 3 zeros of $$z^6+1$$ in the UHP.

### Applications contour integrals

Example 1 (cont): Note that $$z_1 = e^{\pi i/6}$$, $$z_2 = i$$, $$z_3 = e^{5\pi i/6}$$.

Residue Theorem $$\Rightarrow$$ $\int_C f= 2 \pi i \sum_{j=1}^{3}\underset{z=z_j}{\res} f(z)\qquad (1)$

Also note $$f$$ has the form $$p/q$$: for each $$z_j$$, there holds: $p(z_i)\neq 0, q(z_j)=0, q'(z_i)\neq 0.$

### Applications contour integrals

Example 1 (cont): Thus we have simple poles at each $$z_j$$ and so Theorem 3 (Lec 34) implies

$$\text{RHS} = 2 \pi i \ds \sum_{j=1}^{3} \left.\frac{z^2}{(z^6+1)^{\prime}}\right|_{z=z_j}$$ $$= 2 \pi i \ds \sum_{j=1}^{3} \frac{z_j^2}{6z_j^5}$$ $$= 2 \pi i \ds \sum_{j=1}^{3} \frac{1}{6z_j^3}$$

$$= 2 \pi i \left( \ds \frac{1}{6i} -\frac{1}{6i}+ \frac{1}{6i} \right)$$ $$= \dfrac{\pi}{3}.\qquad \qquad \qquad$$

### Applications contour integrals

Example 1 (cont): Note

$$\ds\int_C f = \int_{\Gamma_1}f + \int_{\Gamma_2}f \quad (2)$$

$$\quad(\alpha)$$ $$\qquad (\beta)$$

As $$R\ra \infty$$, $$(\beta) \ra 2I$$.

Claim: $$(\beta) \ra 0$$ as $$R\ra \infty$$.

### Applications contour integrals

Example (cont):

Claim: $$(\beta) \ra 0$$ as $$R\ra \infty$$.

Note $$\big|(\alpha)\big| \leq M_R \ell_R\; \,(*)\,$$ where $$\ell_R = \text{length } \Gamma_1 = \pi R$$

and $$M_R = \ds\max_{z\in \Gamma_1 } \left| \,f(z) \right|$$ $$\leq \ds \max_{|z|=R } \left| \frac{z^2}{z^6+1} \right|$$ $$\leq \ds \frac{R^2}{R^6-1}$$.

(via reverse triangle inequality: $$|a+b|\geq \big||a|-|b|\big|$$).

### Applications contour integrals

Example 1 (cont): So $$(*)$$ implies

$$\big|(\alpha)\big| \leq \ds \frac{\pi R \cdot R^2}{R^6-1}$$ $$=\ds\frac{\pi}{R^3 -\frac{1}{R^3}}$$

this espression $$\ra 0$$ as $$R\ra \infty$$. So $$\ds\lim_{R\ra \infty}$$ in $$(2)$$ $$\Rightarrow$$

$$\ds \frac{\pi}{3} = 2I + 0 \;$$ $$\Rightarrow \; I = \ds\frac{\pi}{6}$$.

### Applications contour integrals

Example 2: Evaluate $$I = \ds \int_{0}^{\infty}\frac{\sin x}{x}dx$$.

Note $$f$$ is even, so if $$I$$ exists

$$I = \ds \frac{1}{2} \int_{-\infty}^{\infty}\frac{\sin x}{x}dx$$ $$= \ds \frac{1}{2} \text{PV}\int_{-\infty}^{\infty}\frac{\sin x}{x}dx$$.

### Applications contour integrals

 Example 2 (cont): Take $$f(z)=\dfrac{e^{iz}}{z}$$. Notice that $$f$$ is analytic on $$\C_*$$, so Cauchy $$\Rightarrow$$ $\int\limits_{\gamma_1 + C_{\rho} + \gamma_2 + C_R} f = 0.$

### Applications contour integrals

Example 2 (cont): Now $$\ds \int_{\gamma_1 } f + \int_{C_{\rho} } f+ \int_{\gamma_2} f+ \int_{ C_R} f = 0 \quad(*)$$

$$\;\qquad I_1\qquad I_2 \qquad I_3 \qquad I_4$$

$$\qquad\qquad I_1 = \ds \int_{-R}^{-\rho} \frac{e^{ix}}{x}dx$$ $$= \ds- \int_{\rho}^{R} \frac{e^{i w}(-1)}{-w}dw$$.

$$\qquad\qquad I_3 = \ds \int_{\rho}^{R} \frac{e^{ix}}{x}dx$$.

### Applications contour integrals

Example 2 (cont): $$\Rightarrow I_1 + I_3 = \ds \int_{\rho}^{R} \frac{e^{ix}- e^{-ix}}{x}dx$$

$$\;\quad\qquad= \ds 2 i \int_{\rho}^{R} \frac{\sin x}{x}dx$$

Need limit as $$\rho \downarrow 0$$ and $$R\ra \infty$$.

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