Complex Analysis MCQs

1

Mathematics • Complex Analysis

Which of the following statements is INCORRECT? Which of the following statements is INCORRECT?

A. The composite of two continuous functions is also continuous. The composite of two continuous functions is also continuous. B. Let f(x + iy) = u(x,y) + iv(x,y). If u and v both have continuous partial derivatives with respect to x and y at (x0,y0), then f is also differentiable at (x0,y0). Let f(x + iy) = u(x,y) + iv(x,y). If u and v both have continuous partial derivatives with respect to x and y at (x0,y0), then f is also differentiable at (x0,y0). C. If f is analytic at all points interior to and on a simple closed contour C, then the integral of f(z) dz over C is 0. If f is analytic at all points interior to and on a simple closed contour C, then the integral of f(z) dz over C is 0. D. If f is a continuous function defined on domain D, then the integral of f(z) dz from z1 to z2, where z1 and z2 lie in D, is the same along all contours connecting z1 and z2 and lying entirely in D. If f is a continuous function defined on domain D, then the integral of f(z) dz from z1 to z2, where z1 and z2 lie in D, is the same along all contours connecting z1 and z2 and lying entirely in D.

सही उत्तर: B Correct Answer: B

Continuity of the first partial derivatives of u and v alone does not ensure complex differentiability; the Cauchy-Riemann equations must also hold at the point. Option B is the answer marked in the source paper.

2

Mathematics • Complex Analysis

Using the Residue Theorem, evaluate the integral of (5z - 2)/(z(z - 1)) dz over the contour C: |z| = 2. Using the Residue Theorem, evaluate the integral of (5z - 2)/(z(z - 1)) dz over the contour C: |z| = 2.

A. 6pi i 6pi i B. 10pi i 10pi i C. 5pi i 5pi i D. 8pi i 8pi i

सही उत्तर: B Correct Answer: B

The contour contains the poles z = 0 and z = 1. Their residues are 2 and 3, whose sum is 5. Thus the integral is 2pi i x 5 = 10pi i.

3

Mathematics • Complex Analysis

If a function f(z), defined on some domain D of the complex plane, is differentiable on D, it is said to be: If a function f(z), defined on some domain D of the complex plane, is differentiable on D, it is said to be:

A. Analytic function Analytic function B. Non-singular function Non-singular function C. Bounded function Bounded function D. Harmonic function Harmonic function

सही उत्तर: A Correct Answer: A

A complex function differentiable at every point of a domain is analytic, or holomorphic, on that domain.

4

Mathematics • Complex Analysis

Let f(r cos(theta) + i r sin(theta)) = u(r,theta) + iv(r,theta). For f to be differentiable at (r,theta), the Cauchy-Riemann equations state that: Let f(r cos(theta) + i r sin(theta)) = u(r,theta) + iv(r,theta). For f to be differentiable at (r,theta), the Cauchy-Riemann equations state that:

A. u_r = -v_r/r, v_r = -u_theta/r u_r = -v_r/r, v_r = -u_theta/r B. u_r = v_theta/r, v_r = -u_theta/r u_r = v_theta/r, v_r = -u_theta/r C. u_r = -v_theta/r, v_theta = -u_theta/r u_r = -v_theta/r, v_theta = -u_theta/r D. u_r = -v_theta/r, v_r = -u_theta/r u_r = -v_theta/r, v_r = -u_theta/r

सही उत्तर: B Correct Answer: B

In polar coordinates the Cauchy-Riemann equations are u_r = v_theta/r and v_r = -u_theta/r.

5

Mathematics • Complex Analysis

If C is a positively oriented simple closed contour within and on which f is analytic, except for a finite number of singular points z_k (k = 1,2,...,n) interior to C, then the Residue Theorem states that: If C is a positively oriented simple closed contour within and on which f is analytic, except for a finite number of singular points z_k (k = 1,2,...,n) interior to C, then the Residue Theorem states that:

A. Integral_C f(z)dz = e pi i (sum_k Res f(z_k)/k!) Integral_C f(z)dz = e pi i (sum_k Res f(z_k)/k!) B. Integral_C f(z)dz = 2pi i (sum_k Res f(z_k)) Integral_C f(z)dz = 2pi i (sum_k Res f(z_k)) C. Integral_C f(z)dz = 2pi i (sum_k Res f(z_k)/k!) Integral_C f(z)dz = 2pi i (sum_k Res f(z_k)/k!) D. Integral_C f(z)dz = sum_k Res f(z_k) Integral_C f(z)dz = sum_k Res f(z_k)

सही उत्तर: B Correct Answer: B

By the Residue Theorem, the contour integral equals 2pi i times the sum of the residues at all singularities inside C.

6

Mathematics • Complex Analysis

Let C denote the boundary of the circle with radius 3 around the centre z_1 = 2 + i. Calculate the integral I = integral_C e^(az)/(z - 2 - i) dz. Let C denote the boundary of the circle with radius 3 around the centre z_1 = 2 + i. Calculate the integral I = integral_C e^(az)/(z - 2 - i) dz.

A. 0 0 B. 2pi i e^(a(2 - i)) 2pi i e^(a(2 - i)) C. 2pi i e^(a(2 + i)) 2pi i e^(a(2 + i)) D. 2pi i e^(a(2 + 2i)) 2pi i e^(a(2 + 2i))

सही उत्तर: C Correct Answer: C

By Cauchy's integral formula, the integral equals 2pi i times e^(az) evaluated at z = 2 + i, giving 2pi i e^(a(2 + i)).

7

Mathematics • Complex Analysis

Compute the integral of cos(z/2) dz from 0 to pi + 2i. Compute the integral of cos(z/2) dz from 0 to pi + 2i.

A. e + 1/e e + 1/e B. sin(pi/4) sin(pi/4) C. 2pi i 2pi i D. 0 0

सही उत्तर: A Correct Answer: A

An antiderivative is 2sin(z/2). At z = pi + 2i this is 2sin(pi/2 + i) = 2cosh(1) = e + 1/e; the lower endpoint contributes zero.

8

Mathematics • Complex Analysis

Select the function f(z) from below that is analytic on the unit disk |z|<1. Select the function f(z) from below that is analytic on the unit disk |z|<1.

A. f(z)=e^x e^(-iy) f(z)=e^x e^(-iy) B. f(z)=cos(x)cosh(y)-i sin(x)sinh(y) f(z)=cos(x)cosh(y)-i sin(x)sinh(y) C. f(z)=3x+i3xy f(z)=3x+i3xy D. f(z)=2x+ixy^2 f(z)=2x+ixy^2

सही उत्तर: B Correct Answer: B

Option B is cos(x+iy)=cos(z), which is entire and therefore analytic on the unit disk.

9

Mathematics • Complex Analysis

The power-series expansion sum of z^(4n+1)/(2n)! for n=0,1,2,... represents which analytic function? The power-series expansion sum of z^(4n+1)/(2n)! for n=0,1,2,... represents which analytic function?

A. z sinh(z) z sinh(z) B. z sinh(z^2) z sinh(z^2) C. 1/z^2 1/z^2 D. z cosh(z^2) z cosh(z^2)

सही उत्तर: D Correct Answer: D

Since cosh(z^2)=sum z^(4n)/(2n)!, multiplying by z gives sum z^(4n+1)/(2n)!.

10

Mathematics • Complex Analysis

The Taylor-series expansion sum of (-1)^n z^(2n+1)/(2n+1)! for n=0,1,2,... represents which analytic function? The Taylor-series expansion sum of (-1)^n z^(2n+1)/(2n+1)! for n=0,1,2,... represents which analytic function?

A. tan(z) tan(z) B. sin(z) sin(z) C. cosh(z) cosh(z) D. cos(z) cos(z)

सही उत्तर: B Correct Answer: B

This is the standard Maclaurin expansion z-z^3/3!+z^5/5!-..., which equals sin(z).

Complex Analysis MCQs

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