Advanced Mathematics

What is Numerical integration formula in simpson's 3/8 rule?

Write Gauss forward and backward interpolation formula.

Write Trapezoidal formula's for integration.

State convolution theorem for fourier transform.

What are the existence condition for Laplace Transform?

Find the Laplace transform of following $F(t)=t{e}^t\sin t$.

Find the inverse laplace transform following function $\frac{4s+5}{(s-1{)}^2(s+2)}$.

State fundamental theorem of finite difference calculus.

Define Lagrange's formulae.

List the properties of mobius transformations.

Use stialing formula to find f(1.22) from following table : | x | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 | 1.7 | 1.8 | |---|---|---|---|---|---|---|---|---|---| | f(x) | .84147 | .89121 | .93204 | .96356 | .98545 | .99749 | .99957 | .97385 | .97385 |

Find inverse Laplace transform of the function. $$\frac{s}{s^4+s^2+1}$$

Find the root of the following equation by Newton - Raphson method. $$3x-\cos x+6=0$$

Find the Laplace transform of a. $t{e}^{-2t}\sin 3t$ b. $\sinh 3t{\cos }^{2}t$

Given : ${\log }_{10}654=2.8156$, ${\log }_{10}658=2.8182$, ${\log }_{10}659=2.8189$, ${\log }_{10}661=2.8202$ use Newton's divided difference formula to find value of ${\log }_{10}656$.

Compute the value of following integral by Trapezoidal rule : - $${\int }_{0.2}^{1.4}(\sin x-{\log }_{e}x+e^x)dx$$

Show that the polar form of C-R equation's is. $$\frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial \theta },\frac{\partial v}{\partial r}=-\frac{1}{r}\frac{\partial u}{\partial \theta }$$ Using this result show that log z is analytic.

Find the bilinear transformation that map's the point 0, -i, -1 in z - plane onto the point's W = i, 1, 0. What are the invariant point's of this transformation?

Find inverse Z transform of f(z) where R.O.C ; $|z|>2$. $$\frac{z^2}{(z+2)(z^2+4)}$$

Prove that $$L\left(\frac{{\sin }^{2}t}{t}\right)=\frac{1}{4}\log \left(\frac{s^2+4}{s^2}\right)$$ and hence deduce the integral ${\int }_0^{\infty }\frac{{\sin }^{2}t}{t^2}dt$.