Advanced Mathematics

Show that $2\mu \delta \equiv (E-E^{-1})$.

Define divided difference.

Write Newton- Raphson iteration formula.

Write Simpson's trapezoidal rule.

Define Fourier sine transform.

State convolution theorem for Laplace transform.

Define Z- transform.

Prove that function u = cosx.coshy is harmonic function.

Define Mobius transformation.

Write Cauchy-Riemann equations.

Prove that $\left(\frac{{\Delta }^2}{E}\right)\left(\frac{e^{x}E{e}^x}{{\Delta }^2{e}^x}\right)=e^x$, h = 1.

Find f(3.5) and f(4) from given data - | x | 3 | 5 | 7 | 9 | 11 | |---|---|---|---|---|---| | f(x) | 6 | 24 | 58 | 108 | 174 |

Find real root of equation $x^3-3x-5=0$ corrected upto 4 decimal place using Newton-Raphson method.

Find inverse Fourier cosine transform of $\frac{1}{1+p^2}$.

If f(t) is a periodic function with period T >0, then prove that - $$L[f(t);p]=\frac{1}{1-e^{-pT}}{\int }_0^T{e}^{-pt}f(t)dt$$

Find Z-transform of $u_n=a^n\cos bn$, n $\ge$ 0.

Show that function $f(z)=u+iv$ where $f(z)=\frac{x^3(1+i)-y^3(1-i)}{x^2+y^2}$, ($z\ne 0$), f(0) = 0 is continuous at origin and satisfies Cauchy-Riemann equations at origin but f '(z) does not exists at origin.

(a) Find value of $\frac{dy}{dx}$ and $\frac{d^{2}y}{d{x}^2}$ at x = 5 from given data. | x | 0 | 2 | 3 | 4 | 7 | 9 | |---|---|---|---|---|---|---| | y=f(x) | 4 | 26 | 58 | 112 | 466 | 922 | (b) Find value of integral ${\int }_0^1\frac{dx}{1+x^2}$ using Simpsons $\frac{1}{3}$ and $\frac{3}{8}$ rule by dividing range into 6 equal parts.

(a) Find a polynomial in powers of (x-3) from following data. | x | 5 | 11 | 27 | 34 | 42 | |---|---|---|---|---|---| | f(x) | 23 | 899 | 17315 | 35606 | 68510 | (b) Use Stirling's interpolation formula to find f(128) from given data. | x | 120 | 125 | 130 | 135 | 140 | |---|---|---|---|---|---| | f(x) | 49225 | 48316 | 47236 | 45926 | 44306 |

(a) Find Laplace transform of the function $\sin \sqrt{t}$ and hence obtain Laplace transform of $\frac{\cos \sqrt{t}}{\sqrt{t}}$. (b) Find inverse Laplace transform of $\frac{p^2-4}{(p^2+1)(p^2+4{)}^2}$.

(a) Find the Fourier transform of $f(t)=\{\begin{array}{ll}1-t^2,& |t|<1\\ 0,& |t|>1\end{array}$ and hence evaluate ${\int }_0^{\infty }\left(\frac{t\cos t-\sin t}{t^3}\right)\cos \frac{t}{2}dt$. (b) Find inverse Z-transform of $f(z)=\frac{z^2}{(z-\frac{1}{4})(z-\frac{1}{5})}$ if region of convergence is (i) $|z|<\frac{1}{5}$ (ii) $\frac{1}{5}<|z|<\frac{1}{4}$

(a) If $u+v=\frac{2\sin 2x}{(e^{2y}+e^{-2y}-2\cos 2x)}$ and f(z) = u + iv is an analytic function then find f(z) in terms of z. (b) Show that transformation $w=\frac{2z+3}{z-4}$ transfers circle $x^2+y^2-4x=0$ into straight line.