Question

3 B 1. Find the third roots of 21+ Find the inverse of the Laplace transform 2. tan" G) 3. Check the existence of the Laplace transform for the given function and hence she that -02:49 in 133+ 4 S- where LF(t)) is represent the place transform of (1) [Hint: 2 cos Acos B = (A-2).sin(A+B) + sin(A - m = sin cos sin(A + B) - Sin(A) = 0 4. Find the Fourier Sine series of f(x) <rci 5. Using the Fourier cosine integral, show that COS ITW cos wx dw- w cos 0. 0<x</2 12/2 Find the Fourier transform of x < 0

Answer 1

let 1+1=r(CosetiSina) lti =(rCos 8) +i(rSin) Equating the real and imaginary parts we have rCos 0= 1 and Sind = 1 Dividing the above two equations tan = = 1 7 A= 4 Andr= x2 + y2 = V1+1 = 2 Thus ग 1'i = x2(cos(+32(0) = vācis Vācis 2kr+ (Since Cise=Cose+i Sin) 4 (Since Cis 8=Cis(2k 7+ = V2cis (8 +1) (18k+14)

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Cube roots of 1+i are 71/3 7T (1+i))"3 = V2cis (8x+1) (Vācis(et =(12)"(cis(ex+1)]" - - [ca(a +913(1)] (By De Moivre's Theorem, (rcis 0)* = *Cis(no) 21/ 6Cis (8k+1) where k = 0,1,2 Plugging in k=0 we get 21/6 Cis (80)+1) 12 =216 Cis (9) Plugging in k=1 we get ग zvoj 216 Cis (8(1)+1) 12 Cie (8 21/6 Cis (9) 37 21/6 Cis Pluggging in k= 2 we get 21/6 Cis (8(2)+1) 12 216 Cis (17) = 177 21 Cis 12 Therefore the three cube roots of unity are 7 37 177 216 Cis ,216 Cis 216 Cis 12 4. 12

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$L^{-1}\left\{\arctan \left(\frac{s}{s+1}\right)\right\}$

$\lim _{s\to \infty \:}\left(\arctan \left(\frac{s}{s+1}\right)\right)$

$\mathrm{Apply\:the \:Limit\: Chain\:Rule}$

= $\lim _{s\to \infty \:}\left(\arctan \left(\frac{s}{s+1}\right)\right)=\frac{\pi }{4}$

$\mathrm{Since\:}\frac{\pi }{4}\ne 0$

$=\mathrm{No\:solution\:in\:terms\:of\:standard\:functions}$