GATE Engineering Mathematics - Complex Variables
Exam Duration: 45 Mins Total Questions : 15
In the Taylor's series expansion of ex about x=2, the coefficient of (x-2)4 is
- (a)
\(\frac { 1 }{ 4! } \)
- (b)
\(\frac { { 2 }^{ 4 } }{ 4! } \)
- (c)
\(\frac { { e }^{ 2 } }{ 4! } \)
- (d)
\(\frac { { e }^{ 4 } }{ 4! } \)
if \(\phi (x,y)and\psi (x,y)\) are functions with continuous second derivative, then
\(\phi (x,y)+\psi (x,y)\)
can be ecpressed as an analytic funtion of \(x+iy=(i=\sqrt { -1) } \)when
- (a)
\(\frac { \partial \phi }{ \partial x } =-\frac { \partial \psi }{ \partial x } ;\frac { \partial \phi }{ \partial y } =\frac { \partial \psi }{ \partial y } \)
- (b)
\(\frac { \partial \phi }{ \partial y } =-\frac { \partial \psi }{ \partial x } ;\frac { \partial \phi }{ \partial x } =\frac { \partial \psi }{ \partial y } \)
- (c)
\(\frac { { \partial }^{ 2 }\phi }{ { \partial x }^{ 2 } } +\frac { { \partial }^{ 2 }\phi }{ { \partial y }^{ 2 } } =\frac { { \partial }^{ 2 }\psi }{ { \partial w }^{ 2 } } +\frac { \partial ^{ 2 }\psi }{ { \partial y }^{ 2 } } =1\)
- (d)
\(\frac { \partial \phi }{ \partial x } +\frac { \partial \phi }{ \partial y } =\frac { \partial \psi }{ \partial x } +\frac { \partial \psi }{ \partial y } =0\)
The \(\lim _{ x\rightarrow 0 }{ \frac { sin\left[ \frac { 2 }{ 3 } x \right] }{ x } } \) is
- (a)
\(\frac { 2 }{ 3 } \)
- (b)
1
- (c)
\(\frac { 1 }{ 4 } \)
- (d)
\(\frac { 1 }{ 2 } \)
The analytic function \(f(z)=\frac { z-1 }{ { z }^{ 2 }+1 } \) has singularities at
- (a)
1 and -1
- (b)
1 and 1
- (c)
1 and -i
- (d)
i and -i
The value of the function f(x)=\(\lim _{ x\rightarrow 0 }{ \frac { { x }^{ 3 }+{ x }^{ 2 } }{ { 2x }^{ 3 }-{ 7x }^{ 2 } } } \) is
- (a)
0
- (b)
\(-\frac { 1 }{ 7 } \)
- (c)
\(\frac { 1 }{ 7 } \)
- (d)
\(\infty \)
\(\lim _{ x\rightarrow \infty }{ \frac { x-sinx }{ x+cosx } } \) equals to
- (a)
1
- (b)
-1
- (c)
\(\infty \)
- (d)
-\(\infty \)
The value of the \(\oint { \frac { -3z+4 }{ { z }^{ 2 }+4z+5 } } dz\) where C is the circle |z|=1 is given by
- (a)
0
- (b)
\(\frac { 1 }{ 10 } \)
- (c)
\(\frac { 4 }{ 5 } \)
- (d)
1
Which of the following functions would have only odd poweers of x in its Taylor series expansion about the point x=0?
- (a)
sin (x3)
- (b)
sin (x3)
- (c)
cos (x3)
- (d)
cos (x2)
The residue of a complex function \(X(z)=\frac { 1-2z }{ z(z-1)(z-2) } \) at its pole are
- (a)
\(\frac { 1 }{ 2 } ,-\frac { 1 }{ 2 } and1\)
- (b)
\(\frac { 1 }{ 2 } ,-\frac { 1 }{ 2 } and-1\)
- (c)
\(\frac { 1 }{ 2 } ,1,and-\frac { 3 }{ 2 } \)
- (d)
\(\frac { 1 }{ 2 } ,-1,and-\frac { 3 }{ 2 } \)
Consider likely applicability of Cauchy's integral theorem to evaluate the following integral counter clockwise around the unit circle C.
\(1=\oint { _{ c } } sec\quad z\quad dz,z\) being a complex variable. The value of I will be
- (a)
l=0,singularity set \(\oint { . } \)
- (b)
l-0, singularity set \(\left\{ \pm \frac { 2n+1 }{ 2 } \pi ,n=0,1,2,... \right\} \)
- (c)
\(l=\frac { \pi }{ 2 } ,singularities\quad set=\left\{ \pm n\pi ;n=0,1,2,... \right\} \)
- (d)
None of the above
The residue of the function \(f(z)=\frac { 1 }{ { (z+2) }^{ 2 }(z-2) } \) at z=2 is
- (a)
\(-\frac { 1 }{ 32 } \)
- (b)
\(-\frac { 1 }{ 16 } \)
- (c)
\(\frac { 1 }{ 16 } \)
- (d)
\(\frac { 1 }{ 32 } \)
The Taylor series expansion of \(\frac { sin\quad x }{ x-\pi } at\quad x=\pi \quad \)is given by
- (a)
\(1+\frac { (x-\pi { ) }^{ 2 } }{ 3! } \)
- (b)
\(-1-\frac { (x-\pi { ) }^{ 2 } }{ 3! } +...\)
- (c)
\(1-\frac { (x-\pi { ) }^{ 2 } }{ 3! } \)
- (d)
\(-1+\frac { (x-\pi { ) }^{ 2 } }{ 3! } +...\)
The value of the contour integral \(\underset { |z-i|=2 }{ \oint { \frac { 1 }{ { z }^{ 2 }+4 } } } dz\) in positive sense is (z-i)=2
- (a)
\(\frac { \sqrt { \pi } }{ 2 } \)
- (b)
\(-\frac { \pi }{ 2 } \)
- (c)
\(-\frac { \sqrt { \pi } }{ 2 } \)
- (d)
\(\frac { \pi }{ 2 } \)
The Cauchy integral therom states if f(z) is analytic in a simple connected domain D, then \(\int { _{ c } } f(z)dz=0\) on every simple closed path C and Dn. The condition of analytically in this therom is
- (a)
neccessary
- (b)
necessary and sufficient
- (c)
sufficient
- (d)
None of the above
The most general complex analytical function f(z)=u(x,y)+iv(x,y) for u =x2-y2 is
- (a)
z
- (b)
2z
- (c)
z2
- (d)
\(\frac { 1 }{ { z }^{ 2 } } \)