This set of Complex Function Theory Multiple Choice Questions & Answers (MCQs) focuses on “Cauchy-Riemann (C-R) Equations”.
1. The function f(z) = xy + iy is continuous everywhere but not differential anywhere.
a) True
b) False
View Answer
Explanation:
Given f(z) = xy +iy
i.e. u=xy, v=y
x and y are continuous everywhere and consequently u(x,y) = xy and v(x,y) = y are continuous everywhere.
Thus, f(z) is continuous everywhere.
But
u=xy | v=y |
---|---|
ux=y | vx=0 |
uy=x | vy=1 |
ux≠vy | uy≠ux |
C-R equations are not satisfied. Hence, f(z) is not differentiable anywhere
though it is continuous everywhere.
2. Which of the following is the derivative of the function f(z) = zn?
a) nzn-1
b) 1
c) 0
d) n2
View Answer
Explanation:
Let z= reiθ
zn=rn(einθ)
zn=rn(cos nθ + isinnθ)
i.e., u= rncosnθ, v= rnsinnθ
\( \frac{∂u}{∂r}=nr^{n-1}cosnθ, \frac{∂v}{∂r} = nr^{n-1}sinnθ \)
\( \frac{∂u}{∂θ}=-nr^nsinnθ, \frac{∂v}{∂θ}=nr^ncosnθ \)
\( \frac{∂u}{∂r}=\frac{1}{r}\frac{∂v}{∂θ}, \frac{∂v}{∂r}=-\frac{1}{r}\frac{∂u}{∂θ}\)
C-R equations are satisfied and the partial derivatives are continuous.
Hence,the function is analytic.
\( f'(z)=\frac{u_r+iv_r}{e^{iθ}} = \frac{nr^{n-1}cosnθ+in^{r-1}sinnθ}{e^{iθ}} \)
\( =\frac{nr^ncosnθ+inr^nsinnθ}{re^{iθ}} = \frac{n[r(cosnθ+isinθ)]^n}{re^{iθ}} \)
\( = \frac{n[re^{iθ}]^n}{re^{iθ}} = \frac{nz^n}{z} = nz^{n-1} \)
3. Which of the following is the derivative of the function f(z) = 1/z?
a) \( \frac{-1}{z^2} \)
b) \( \frac{1}{z^3} \)
c) \( \frac{-1}{z^3} \)
d) \( \frac{1}{z^2} \)
View Answer
Explanation:
Let z=reiθ
\( f(z) = \frac{1}{z}=\frac{1}{re^{iθ}}=\frac{1}{r}e^{-iθ} \)
\( \frac{1}{r}[cosθ-isinθ] \)
At z=0,r=0 and so f(z) is not defined at z=0
C-R equations are
\( \frac{∂u}{∂r}=\frac{1}{r}\frac{∂v}{∂θ}, \frac{∂u}{∂θ}=-r\frac{∂v}{∂r} \)
\( u=\frac{cosθ}{r} v=\frac{-sinθ}{r} \)
\( u_r=\frac{-cosθ}{r^2} v=\frac{sinθ}{r^2} \)
\( u_θ=\frac{-sinθ}{r} v_θ=\frac{-cosθ}{r} \)
C-R equations are satisfied.
Hence,f(z)is analytic everywhere except z=0.
f(z)=u+iv
Differentiating w.r.t. r’,
\( e^{iθ} f'(z) = u_r + iv_r \)
\( f'(z) = \frac{u_r+iv_r}{e^{iθ}}=\frac{-cosθ+isinθ}{r^2e^{iθ}}=\frac{-[cosθ-isinθ]}{r^2e^{iθ}} \)
\( =\frac{-e^{iθ}}{r^2e^{iθ}}=\frac{-1}{(re^iθ)^2}=\frac{-1}{z^2} \)
4. Which of the following co-ordinates is correct on which the function f(z)= √(|xy|) is not regular, although the Cauchy – Riemann equations are satisfied?
a) (0, 0)
b) (1, 2)
c) (-1, -2)
d) (2, 1)
View Answer
Explanation:
\(f(z)= u(x,y)+i v(x,y)= \sqrt{|xy|} \)
\( i.e., u(x,y)= \sqrt{|xy|}, v(x,y)=0 \)
\( u_x = (\frac{∂u}{∂x})_{(0,0)} = lim_{∆x→0}\frac{0-0}{∆y} = 0 \)
\( u_y (0,0)= lim_{∆y→0}\frac{0-0}{∆y}=0 \)
\( v_x (0,0)= lim_{∆x→0}\frac{0-0}{∆x}=0 \)
\( v_y (0,0)= lim_{∆y→0}\frac{0-0}{∆y}=0 \)
Clearly, ux = vy and uy = -vx at the origin
\( Now, lim_{∆z→0}\frac{f(0+∆z)-f(0)}{∆z}= lim_{∆z→0}\frac{\sqrt{|∆x∆y|}-0}{∆x+i∆y} \)
\( =lim_{∆y}= m∆x\frac{\sqrt{|m||∆x|^2}}{∆x(1+im)} = \frac{\sqrt{|m|}}{1+im} \)
The limit is not unique, since it depends on m. Therefore, f'(0) does not exist.
Hence, f(z) is not regular at the origin.
5. Which of the following is not an analytic function?
a) 2xy + i(x2-y2)
b) xy
c) x + y
d) 2xy
View Answer
Explanation:
Let f(z) = 2xy + i(x
u=2xy | v=x2-y2 |
---|---|
\(\frac{∂u}{∂x}=2y\) | \(\frac{∂v}{∂x}=2x\) |
\(\frac{∂u}{∂y}=2x\) | \(\frac{∂v}{∂y}=-2y\) |
ux≠vy and uy≠-vx
Hence, C-R equations are not satisfied.
Hence, f(z) is not an analytic function.
6. Which of the following co-ordinates is correct on which the function f(z) = x2+iy3 is not regular, although the Cauchy-Riemann equations are satisfied?
a) (0,0)
b) (1, 2)
c) (2,1)
d) (-1,-2)
View Answer
Explanation:
Let f(z) = x3 +iy3
i.e., u+iv=x2+iy3
u=x2 | v=y3 |
---|---|
ux=2x | vx=0 |
uy=0 | vy=3y2 |
ux≠vy and uy= -vx
C-R equations are not satisfied.
Hence, f(z) is not analytic.
2x=3y2 is true at x=0, y=0
Hence, C-R equations are satisfied at (0,0)
7. Which of the following curves is correct on which the function f(z)= (x-y)2+ 2i (x+y) is analytic and C-R equations are satisfied?
a) x-y=1
b) x+y=1
c) x2+y2=1
d) x2-y2=1
View Answer
Explanation:
Given: f(z)=(x-y)2+ 2i(x+y)
i.e., u+iv=(x-y)2+ i2(x+y)
u=(x-y)2 | v=2(x+y) |
---|---|
ux=2(x-y) | vx=2 |
uy=2(x-y)(-1)=-2(x-y) | vy=2 |
ux= vy, uy=-vx
2(x-y)= 2, -2(x-y)= -2
x-y=1, x-y=1
Hence, C-R equations are satisfied only if x-y=1.
8. Which of the following is the derivative of f(z)=cos hz?
a) sin hz
b) cosec hz
c) sec hz
d) tan hz
View Answer
Explanation:
\( Given: f(z)= \cos(hz) = \cos(iz)=cos([i(x+iy)]) \)
\( =\cos(ix-y)=\cos(ix)\cos(y)+\sin(ix)\sin(y) \)
\( u+iv=\cos(hx)\cos(y)+i\sin(hx)\sin(y) \)
\(u=\cos(hx)cos(y)\) | \(v=\sin(hx)sin(y)\) |
---|---|
\(u_x=\sin(hx)cos(y)\) | \(v_x=\cos(hx)sin(y)\) |
\(u_y=-\cos(hx)sin(y)\) | \(v_y=\sin(hx)cos(y)\) |
ux,uy, vx,vy exist and are continuous.
ux=vy and uy=-vx
C-R equations are satisfied
f(z)is analytic everywhere.
\( Now, f'(z) = u_x+iv_x = \sin(hx)\cos(y) + i\cos(hx)\sin(y) = \sin(h(x+y)) =\sin(hz) \)
9. Which of the following is analytic if u+iv is also analytic?
a) v-iu and-v+iu
b) u+iv and u-iv
c) v+iu and v-iu
d) -v+iu and u+iv
View Answer
Explanation:
Given: u+iv is analytic
C-R equations are satisfied.
i.e., ux= vy and uy= -vx
Since the derivatives of u and v exist therefore it is continuous.
Now, to prove v-iu and -v+iu are also analytic, we should prove that
(i) vx= -uy and vy= ux and
(ii) -vx=uy and vy= ux
(iii) ux,uy,vx and vy are continuous. Since the derivatives of u and v exists from (1) and (2),the derivatives of u and v should be continuous.
Hence, the result (iii) follows.
10. Which of the following is true ifw=f(z) and z=x+iy?
a) \( \frac{dw}{dz}= \frac{∂w}{∂x}=-i\frac{∂w}{∂y} \)
b) \( \frac{dw}{dz}= \frac{∂w}{∂x}= i\frac{∂w}{∂y} \)
c) \( \frac{dw}{dz}= \frac{∂w}{∂x}= \frac{∂w}{∂y} \)
d) \( \frac{dw}{dz}= \frac{∂w}{∂x}= -2\frac{∂w}{∂y} \)
View Answer
Explanation:
Let w=u(x,y)+ iv(x,y)
As f(z) is analytic, we have ux=vy,uy= -vx
\( Now, \frac{dw}{dz}=f'(z)= u_x+iv_x= v_y-iu_y=-i(u_y+iv_y) \)
\( = \frac{∂u}{∂x}+i\frac{∂v}{∂x}= -i[\frac{∂u}{∂y}+i\frac{∂v}{∂y}] \)
\( = \frac{∂}{∂x}(u+iv)= -i\frac{∂}{∂y}(u+iv) \)
\( = \frac{∂w}{∂x}=-i\frac{∂w}{∂y} \)
11. Which of the following are the values of a, b, c if f(z)=(x+ay)+i(bx+cy) is analytic?
a) a=b and c=0
b) a= -b and c=1
c) a=-b and c=0
d) a= -b and c=0
View Answer
Explanation:
If f(z)= u(x,y)+ iv(x,y) is analytic, then f(z)= u(z,0)+ iv(z,0)
Hence, f(z)= z+ibz [x=z and y=0] =(x+iy)+ib(x+iy)
=x+ibx+iy-by
Given: f(z)= x+ay+i(bx+c)
i.e., x+ay+i(bx+cy) = x+ibx+iy-by
By Equating, we get
a=-b, c=1
12. For what values of z, the function w defined by z= e(-v)(cos u + i sin u) ceases to be analytic?
a) 0
b) 1
c) 2
d) 3
View Answer
Explanation:
The function w=f(z) will cease to be analytic at points, where \( \frac{dz}{dw}=0 \)
\(z=e^{-v} e^{iu}= e^{i(u+iv)} \)
\( z= e^{iw} \)
\(\frac{dz}{dw}=ie^{iw}\) =iz = 0 when z=0
Thus, w ceases to be analytic at z=0.
13. For what values of z, the function w defined by \( \frac{z}{(z^2-1)}\) ceases to be analytic?
a) ±1
b) 0
c) 2
d) -2
View Answer
Explanation:
\(Let f(z)= \frac{z}{z^2-1} \)
\(f'(z)= \frac{(z^2-1)(1)- z(2z)}{(z^2-1)^2} = \frac{-(z^2+1)}{(z^2-1)^2} \)
f(z) is not analytic, where f^’ (z)does not exist
i.e.,f’ (z)→∞
i.e., (z2-1)2=0
i.e.,z2-1=0
z= ±1
14. For what values of z, the function w defined by \(\frac{z+i}{(z-i)^2}\) ceases to be analytic?
a) i
b) 0
c) 1
d) 2
View Answer
Explanation:
\(Let f(z)= \frac{z+i}{(z-i)^2} \)
\(f'(z)= \frac{(z-i)^2(1)-(z+i)[2(z-i)]}{(z-i)^4} = \frac{-(z+3i)}{(z-i)^3} \)
f'(z)→ ∞, at z=i
f(z)is not analytic at z=i
15. For what values of z, the function w defined by tan2z ceases to be analytic?
a) \(\frac{(2n-1)π}{2}\),n=1,2,3,…
b) nπ,n=1,2,3,…
c) (n-1)π,n=1,2,3,…
d) 2nπ,n=1,2,3,…
View Answer
Explanation:
\(Let f(z)= \tan^2(z) \)
\(f'(z)=2\tan(z)\sec^2(z)= \frac{2sinz}{\cos^3(z)} \)
\(f'(z)→∞, at \cos^3(z)=0 \)
\(i.e., z=\frac{(2n-1)}{2}π \)
∴f(z) is not analytic at \(z= \frac{(2n-1)π}{2},n=1,2,3,… \)
Sanfoundry Global Education & Learning Series – Complex Function Theory.
To practice all areas of Complex Function Theory, here is complete set of 1000+ Multiple Choice Questions and Answers.
If you find a mistake in question / option / answer, kindly take a screenshot and email to [email protected]
- Practice Numerical Methods MCQ
- Practice Probability and Statistics MCQ
- Check Engineering Mathematics Books
- Apply for 1st Year Engineering Internship