Applied Mathematics - 3 - December 2015

MU Civil Engineering (Semester 3)

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) Find Laplace transform of tsin³t. 5 marks

1 (b) Find half range sine series in (0, π) for x(π-x). 5 marks

1 (c) Find the image of the rectangular region bounded by
x=0, x=3, y=0, y=2 under the transformation ω=z+(1+i). 5 marks

1 (d) Evaluate ∫f(z)dz along the parabola y=2x², z=0 to z=3+18i where f(z)=x²-2iy. 5 marks

2 (a) Find two Laurent's series of $$f\left(z\right)=\frac{1}{z^3(z-1)(z+2)}$$<script type="math/tex; mode=display" id="MathJax-Element-1"> f(z) = \dfrac {1}{z^3 (z-1)(z+2)} </script> about z=0 for
i) |z|<1     ii) 1<|z|<2. 5 marks

2 (b) Find complex form of Fourier series for f(x)=cos h2x+sin h2x in (-2,2). 5 marks

2 (c) Find bilinear transformation that maps 0,1,&inft;; of the z plane into -5, -1, 3 of ω plane. 5 marks