Advanced Engineering Mathematics II

What do you mean by an "Analytic Function".

Define the term "Conformal Mapping".

State Cauchy’s residue theorem.

State maximum - Modulus theorem.

Determine the poles of the function $f(z)=\frac{z^2}{(z-1{)}^2(z+2)}$.

Find the residue of the function $f(z)=\frac{\sin z}{z\cos z}$ at $z=0$.

Write an expression of the generating function for $P_n(x)$.

State orthogonal property of Bessel’s function.

Define ‘Linear Independence’ of vectors.

Explain the term "Minimal Polynomial".

Prove that the function $u=x^3-3x{y}^2-3x^2-3y^2+1$ satisfies Laplace’s equation and determine the corresponding analytic function $u+iv$.

Obtain the Laurent’s series for the function $f(z)=\frac{1}{(4z-z^2)}$ in the region $0<|z|<4$.

Prove the following: $(2n+1)P_n=P_{n+1}^{\prime }-P_{n-1}^{\prime }$

Prove the following: $2J_n^{\prime }(x)=J_{n-1}(x)-J_{n+1}(x)$

Define the term "Inner Product Spaces".

Evaluate ${\int }_C\frac{e^{2z}}{(z-1)(z-2)}dz$ where C is the circle $|z|=3$.

Let V be an inner product space and $\alpha ,\beta$ be vectors in V. Prove that $\alpha =\beta$ if and only if $⟨\alpha ,\gamma ⟩=⟨\beta ,\gamma ⟩$ for every $\gamma$ in V.

Prove that the function $f(z)=\sqrt{|xy|}$ is not analytic at the origin, although the Cauchy-Riemann equations are satisfied at that point.

Using Cauchy’s integral formula, evaluate the integral ${\int }_C\frac{\sin (\pi z^2)+\cos (\pi z^2)}{(z-1)(z-2)}dz$ where C is the circle $|z|=3$.

Evaluate the following integral by contour integration: $${\int }_0^{2\pi }\frac{1}{(5-3\sin \theta {)}^2}d\theta$$

Prove that, when n is a positive integer, $J_n(x)$ is the coefficient of $z^n$ in the expansion of $e^{\frac{x}{2}(z-\frac{1}{z})}$ in ascending and descending powers of z.

Prove that the vectors $(1,2,1)$, $(2,1,0)$, $(1,-1,2)$ form a basis of $R^3$.