Question 1

Write the polar form of Cauchy Riemann equations.

Accepted Answer

In polar form z = r*e^(i*theta), the Cauchy-Riemann equations are: r * du/dr = dv/d(theta) and dv/dr = -(1/r) * du/d(theta).

Question 2

Define analytic function.

Accepted Answer

A function f(z) is analytic at a point z0 if it is complex-differentiable at z0 and at every point in some neighbourhood of z0; the Cauchy-Riemann equations must be satisfied with continuous partial derivatives.

Question 3

State the maximum modulus theorem.

Accepted Answer

The Maximum Modulus Theorem states that if f(z) is analytic and non-constant in a domain D, then |f(z)| attains its maximum only on the boundary of D, never at an interior point.

Question 4

Define zeros of analytic function.

Accepted Answer

A zero of an analytic function f(z) is a point z0 where f(z0) = 0. It is a zero of order m if f(z0) = f'(z0) = ... = f^(m-1)(z0) = 0 but f^(m)(z0) ≠ 0.

Question 5

Define the Removable singularity.

Accepted Answer

A removable singularity is an isolated singular point z0 of f(z) where the limit of f(z) as z -> z0 exists finitely; the singularity can be removed by redefining f(z0) as that limit.

Question 6

Write the Jordan lemma for complex integration.

Accepted Answer

Jordan's Lemma states that if f(z) -> 0 uniformly as |z| -> infinity on the upper semicircle, then the integral of f(z)*e^(iaz) over the semicircular arc tends to zero as R -> infinity, for a > 0.

Question 7

Write the Rodriglue's formula for Legendre function.

Accepted Answer

Rodrigues' formula for Legendre polynomials is Pn(x) = (1/(2^n * n!)) * d^n/dx^n [(x^2 - 1)^n].

Question 8

Define Bessel's function of first and second kind.

Accepted Answer

Bessel function of the first kind Jn(x) is defined by the series solution of Bessel's equation regular at x = 0; the second kind Yn(x) is the linearly independent solution with a logarithmic singularity at x = 0.

Question 9

Define vector subspace.

Accepted Answer

A vector subspace W of a vector space V is a non-empty subset of V that is itself a vector space under the same operations, equivalently W is closed under addition and scalar multiplication.

Question 10

Define Inner product space.

Accepted Answer

An inner product space is a vector space V equipped with an inner product <u, v> satisfying conjugate symmetry, linearity in the first argument, and positive-definiteness.

Question 11

Prove that the function $u = x^3 - 3xy^2 + 3x^2 - 3y^2 + 1$ satisfies Laplace…

Accepted Answer

u = x^3 - 3xy^2 + 3x^2 - 3y^2 + 1 satisfies Laplace's equation. The corresponding analytic function is f(z) = z^3 + 3z^2 + 1 + iC.

Question 12

Show that the transformation $w = \frac{2z + 3}{z - 4}$ changes the circle $x…

Accepted Answer

Under w = (2z+3)/(z-4), the circle x^2 + y^2 - 4x = 0 (which passes through z = 4) maps to the straight line 4u + 3 = 0 because z = 4 is a pole mapped to infinity.

Question 13

Evaluate $\int_c \frac{e^{2z}}{(z+1)^4} dz$, where c is a circle $|z| = 3$.

Accepted Answer

Using Cauchy's integral formula for higher-order poles, the integral of e^(2z)/(z+1)^4 around |z| = 3 equals (4/3)*pi*i*e^(-2).

Question 14

Expand in the series, the function $f(z) = \frac{1}{z^2 - 3z + 2}$ in the reg…

Accepted Answer

f(z) = 1/(z^2 - 3z + 2) = 1/[(z-1)(z-2)] is expanded as a Taylor series for |z| < 1, a Laurent series for 1 < |z| < 2, and a Laurent series for |z| > 2.

Question 15

Show that $\int_0^{2\pi} \frac{d	heta}{2 + \cos 	heta} = \frac{2\pi}{\sqrt{…

Accepted Answer

Using the substitution z = e^(i*theta) to convert the real integral to a complex contour integral, the residue theorem gives the value 2*pi/sqrt(3).

Question 16

Show that when n is a positive integer $J_{-n}(x) = (-1)^n J_n(x)$.

Accepted Answer

Using the series definition of Jn(x) with negative integer n, the relation J_{-n}(x) = (-1)^n * Jn(x) follows from the property of the Gamma function at negative integers.

Question 17

Prove that the set $s \{(1, 2, 1)); (2, 1, 0); (1, -1, 2)$ forms a basis of t…

Accepted Answer

The set {(1,2,1), (2,1,0), (1,-1,2)} forms a basis of V3(R) since the determinant of the matrix with these vectors as rows is -9, which is non-zero, confirming linear independence.

Question 18

Prove that the function $f(z) = u + iv$; where

$$f(z) = \frac{x^2 y^5 (x + i…

Accepted Answer

The function f(z) = x^2*y^5*(x+iy)/(x^4+y^10) satisfies C-R equations at origin by direct computation, but f'(0) does not exist because the limit depends on the path of approach.

Question 19

Use Cauchy's Residue theorem to evaluate:

$$\int_c \frac{\sin \pi z^2 + \cos…

Accepted Answer

Using Cauchy's Residue Theorem, the integral equals 2*pi*i times the sum of residues at z=1 (order 2 pole) and z=2 (simple pole) inside |z|=3.

Question 20

Show by contour integration!

$$\int_{-\infty}^\infty \frac{dx}{(x^2 + 1)^3} …

Accepted Answer

Using the upper-semicircle contour and the residue at the pole of order 3 at z = i inside the contour, the integral evaluates to 3*pi/8.

Question 21

Establish the following result for orthogonality of Legendre Polynomial

$$\i…

Accepted Answer

Orthogonality of Legendre polynomials (integral of Pm*Pn from -1 to 1 = 0 for m != n) is established by showing Pm and Pn satisfy Legendre's equation and using Green's identity (Sturm-Liouville theory).

Question 22

Apply the Gram-Schmidt process to the vectors $\beta_1 = (1, 0, 1)$, $\beta_2…

Accepted Answer

Applying the Gram-Schmidt process to beta1=(1,0,1), beta2=(1,0,-1), beta3=(0,3,4) gives the orthonormal basis: e1=(1/sqrt(2),0,1/sqrt(2)), e2=(1/sqrt(2),0,-1/sqrt(2)), e3=(0,1,0).

Advanced Engineering Mathematics II