Let f(z) be analytic inside and on a simple closed curve C except for a finite number of poles inside C. Suppose that f(z)≠0 on C. If N and P are, respectively, the number of zeros and poles of f(z) inside C, counting multiplicities, prove that 1/2πi ∫C f′(z)/f(z)dz = N - P

Question

Let f(z) be analytic inside and on a simple closed curve C except for a finite number of poles inside C. Suppose that f(z)≠0 on C. If N and P are, respectively, the number of zeros and poles of f(z) inside C, counting multiplicities, prove that

[latex]\frac{1}{2 \pi i} \oint_C \frac{f^{\prime}(z)}{f(z)} d z=N-P[/latex]

Accepted Answer

Let [latex]\alpha_1, \alpha_2, \ldots, \alpha_j[/latex] and [latex]\beta_1, \beta_2, \ldots, \beta_k[/latex] be the respective poles and zeros of f(z) lying inside C [Fig. 5-8] and suppose their multiplicities are [latex]p_1, p_2, \ldots, p_j[/latex] and [latex]n_1, n_2, \ldots, n_k[/latex].

Enclose each pole and zero by non-overlapping circles [latex]C_1, C_2, \ldots, C_j[/latex] and [latex]\Gamma_1, \Gamma_2, \ldots, \Gamma_k[/latex]. This can always be done since the poles and zeros are isolated.
Then, we have, using the results of Problem 5.16,

[latex]\begin{aligned}\frac{1}{2 \pi i} \oint_C \frac{f^{\prime}(z)}{f(z)} d z &=\sum_{r=1}^j \frac{1}{2 \pi i} \oint_{\Gamma_r} \frac{f^{\prime}(z)}{f(z)} d z+\sum_{r=1}^k \frac{1}{2 \pi i} \oint_{C_r} \frac{f^{\prime}(z)}{f(z)} d z \&=\sum_{r=1}^j n_r-\sum_{r=1}^k p_r \&=N-P\end{aligned}[/latex]

Question 5.17: Let f(z) be analytic inside and on a simple closed curve C e...

Related Answered Questions

Evaluate ∫C e^z /(z² + p²)² dz where C is the circle |z|= 4. ...

Verified Answer:

Suppose f(z) = u(x, y) + iv(x, y) is analytic in a region R. Prove that u and v are harmonic in R. ...

Verified Answer:

(a) Let F(z) be analytic inside and on a simple closed curve C except for a pole of order m at z = a inside C. Prove that 1/2πi ∫C F(z) dz = lim z→a 1/(m – 1)! d^m-1 /dz^m-1 {(z – a) ^mF(z)} (b) How would you modify the result in (a) if more than one pole were inside C? ...

Verified Answer:

Let f(x) ={ x² sin(1/x) x ≠ 0 0 x = 0 where x is real. Show that the function f(x) (a) has a first derivative at all values of x for which 0 ≤ x ≤ 1 but (b) does not have a second derivative in 0 ≤ x ≤ 1. (c) Reconcile these conclusions with the result of Problem 5.4. ...

Verified Answer:

Prove Schwarz’s theorem: Let f(z) be analytic for |z| ≤ R, f(0) = 0, and |f(z)| ≤M. Then |f(z)| ≤ M|z|/R ...

Verified Answer:

Let f(z) be analytic in a region R bounded by two concentric circles C1 and C2 and on the boundary [Fig. 5-11]. Prove that, if z0 is any point in R, then f(z0) = 1 /2πi ∫C1 f(z)/ z – z0 dz – 1 /2πi ∫C2 f(z)/z – z0 dz ...

Verified Answer:

Prove that, for n = 1, 2, 3, … , ∫^2π 0 cos^2n θ dθ = 1 . 3 . 5 … (2n – 1)/2 . 4 . 6… (2n) 2π ...

Verified Answer:

Derive Poisson’s formulas for the half plane [see page 146]. ...

Verified Answer:

Verified Answer:

Prove that all the roots of z^7 – 5z³ + 12 = 0 lie between the circles |z| = 1 and |z| = 2. ...

Verified Answer: