Lecture 36
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.
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 \(\theta\in [0,2\pi]\).
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
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 \[\ds \lim_{R\ra \infty } \int_{\Gamma_R } e^{i\alpha z} f(z)\, dz = 0,\; \forall \alpha\gt 0.\]
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 \(C \Rightarrow p=8\).
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).
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 \(0\)).