GTU Complex Variable & Numerical Methods - December 2015 Exam Question Paper

GTU Mechanical Engineering (Semester 4)
Complex Variable & Numerical Methods
December 2015

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a)i Sketch the region

1 < | z + i | \leq 2

$1<\left | z+i \right |\leq 2$ and check whether the region is domain or not?

3 M

1(a)ii Find all the roots of the equation

log z = \frac{i π}{2}

$\log z=\dfrac{i\pi}{2}$ .

2 M

1(a)iii Evaluate

\int_{c} \frac{d z}{z^{2}}

$\int _c \dfrac{dz}{z^2}$ , C is along a unit circle.

2 M

1(b)i Obtain the Taylor's series f(z) = sin z in power of

(z - \frac{π}{4})

$\left ( z-\dfrac{\pi}{4} \right )$

3 M

1(b)ii Find the Laurent's expansion of

\frac{sin z}{z^{3}}

$\frac{\sin z}{z^3}$ at z = 0 and classify the singular point z = 0.

2 M

1(b)iii Is

f (z) = {\sqrt{r e}}^{\frac{i θ}{2}}

$f(z)=\sqrt{re}^{\dfrac{i\theta}{2}}$ analytic? ( r>0, -π < \theta < π)

2 M

2(a) Show that for the function

$f(z)=\left\{\begin{matrix} \dfrac{z^{-2}}{z}; & z\neq 0\\ 0\ ; & z=0 \end{matrix}\right.$ Is not differentiable at z = 0 even though Cauchy Riemann equation are satisfied at z = 0 .

7 M

Solved any one question from Q.2(b) & Q.2(c)

2(b)i Discuss the convergence of the series

$\sum \frac{(2n)!(z-3i)^n}{(n!)^2}$

4 M

2(b)ii Discuss continuity of

$f(z)=\left\{\begin{matrix} \dfrac{Re(z^2)}{|z|^2} ;& z\neq 0\\ 0; & z=0 \end{matrix}\right.$ at z = 0.

3 M

2(c)i State De-Moivre's formula .Find all the root of

$(8i)^{\dfrac{1}{3}}$ in the complex plane.

4 M

2(c)ii Evaluate

$\int _c (x^2-iy^2)dz$ along the parabola y = 2x² from (1, 2) to (2, 8).

3 M

Solved any one question from Q.3 & Q.4

3(a) Using Residue theorem, evaluate

$\int ^{2\pi}_0\dfrac{d\theta}{5-3\sin \theta}$

7 M

3(b)i Evaluate

$\int _c \dfrac{1+z^2}{1-z^2}dz$ , where c is unit circle centred at
(1) z = -1
(2) z = i

4 M

3(b)ii Find the image in the w - plane of the circle |z-3|=2 in the z- plane under the inversion mapping

$w=\dfrac{1}{z}$ .

3 M

4(a)i Show that u(x, y) = x² - y² is harmonic in some domain and find the harmonic conjugate v ( x , y ) .

4 M

4(a)ii Find the Bilinear transformation which maps z = 1 , i , - 1 into ω = 2, i, -2.

3 M

4(b)i Determine the poles of the function

$f(z)=\dfrac{z^2}{(z-1)^2(z+2)}$ and residue at each pole. Evaluate

$\int _c f(z)dz$ , where c is the circle |z| = 3.

4 M

4(b)ii Evaluate

$\int _{c:|z|=2}\dfrac{dz}{z^3(z+4)}$ .

3 M

Solved any one question from Q.5 & Q.6

5(a) Find the Langrage's interpolation polynomial from the following Data :

x	0	1	4	5
f(x)	1	3	24	39

Also find f(2).

7 M

5(b)i Using partial pivoting solve the system of equation by Gauss Elimination method :
x+y+z=7; 3x+3y+4z=24; 2x+y+3z=16

4 M

5(b)ii Find a root of the equation x³ -4x-9=0 using False-position method correct up to three decimal.

3 M

6(a)i Using Newton's Backward interpolation formula , evaluate f (300) from the given table:

x	50	100	150	200	250
y(x	618	724	805	906	1032

4 M

6(a)ii Find real root of x³ 5x + 3 = 0 correct up to three decimal using Newton-Raphson method.

3 M

6(b)i Solve the system of equation by Gauss Seidel method 10x+y+z = 6; x+10y+z = 6; x+y+10z = 6

4 M

6(b)ii Using Newton's Divided difference formula find f (3) from the following table

x	-1	2	4	5
f(x)	-5	13	255	625

3 M

7(a)i Using the power method find the largest Eigen value for the matrix

$A=\begin{bmatrix} 1 & -3 & 2\\ 4 & 4 & -1\\ 6 & 3 & 5 \end{bmatrix}$

7 M

7(b)i Use Runge-Kutta method of second order to find the appropriate value of y (0.2) given that

$\dfrac{dy}{dx}=x-y^2$ ; y(0) and h=0.1

4 M

7(b)ii Evaluate

$\int ^1_0 \dfrac{dx}{1+x^2}$ using Trapezoidal rule taking

$h=\dfrac{1}{5}$

3 M

8(a)i Given

$\dfrac{dy}{dx}=\dfrac{y-x}{y+x}$ with initial condition y = 1 at x = 0 ; find y for x = 1.0 and h = 0.25 by Euler's method

4 M

8(a)ii Using Gauss Forward interpolation formula , evaluate f (55) form the given table :

x	40	50	60	70
f(x)	836	682	436	272

3 M

8(b)i Evaluate

$\int ^1_0\dfrac{dt}{1+t}$ by Gaussian formula for n = 2 and n = 3.

4 M

8(b)ii Prove that E = e^hd.

3 M

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