Update Math416_L25.md

pages/Math416/Math416_L25.md

@@ -93,3 +93,56 @@ $$
 $$
 
 QED
+
+## Application ot valuating definite integrals
+
+Idea:
+
+It is easy to evaluate intervals around closed contours.
+
+Choose contour so one side (where you want to integrate).
+
+Handle the other side by:
+
+- Symmetry
+- length * supremum of absolute value of integrand
+- Bound function by another function whose integral goes to zero.
+
+Example:
+
+Evaluate $\int_0^\infty \frac{\sin x}{x}dx$.
+
+On the contour $\gamma(t)$ be the semicircle in the upper half plane removed the origin.
+
+Then let $f(z)=\frac{e^{iz}}{z}=\frac{\cos z+i\sin z}{z}$, by the Cauchy's theorem,
+
+$$
+\int_\gamma f(z)dz=0
+$$
+
+So $\frac{\sin z}{z}=0$ on $\gamma$.
+
+If $x\in \mathbb{R}$, $f(x)=\frac{e^{ix}}{x}=\frac{\cos x+i\sin x}{x}$.
+
+On the real axis, 
+
+$$
+\begin{aligned}  
+\int_{-R}^{-\epsilon}+\int_\epsilon^R f(x)dx&=\int_{-R}^{-\epsilon}\frac{e^{ix}}{x}dx+\int_\epsilon^R \frac{e^{ix}}{x}dx\\
+&=\int_{-R}^{-\epsilon}\frac{\cos x+i\sin x}{x}dx+\int_\epsilon^R \frac{\cos x+i\sin x}{x}dx\\
+&=\int_{-R}^{-\epsilon}\frac{\cos x}{x}dx+i\int_{-R}^{-\epsilon}\frac{\sin x}{x}dx+\int_\epsilon^R \frac{\cos x}{x}dx+i\int_\epsilon^R \frac{\sin x}{x}dx\\
+&=2i\int_0^\infty \frac{\sin x}{x}dx
+\end{aligned}
+$$
+
+For the clockwise semi-circle around the origin,
+
+$$
+\int_{S_\epsilon} f(z)dz=\int_{S_\epsilon}\frac{e^{iz}}{z}dz
+$$
+
+let $\gamma(t)=\epsilon e^{-it}$, $t\in[-\pi,0]$.
+
+Then $\gamma'(t)=-i\epsilon e^{-it}$,
+
+CONTINUE NEXT TIME.