Mathematical Physics

(i) The value of J_n (x) is:

(ii) The value of H_n(x) is:

(iii) The Laplace transform of 3t – 5 is:

(iv) The Fourier cosine transform of function f(x) = sin ax:

(v) If r₁ and r₂ are two variables, then the symmetry properties of Green's function is:

(vi) The poisson's equation ∇²φ = ρ is:

(vii) The analytic function f(z) of which the real part is e^x cos y is:

(viii) The function f(z) = |z| is:

(ix) In spherical polar coordinates, the value of scalar factors h_r, h_θ, and h_φ are:

(x) Inverse Laplace transform of 1 / (p² + a²) is:

Obtain Rodrigue's formula for Legendre Polynomials.

Prove the recurrence formula for J_n(x).

x J_n (x) = nJ_n (x) – x J_n+1 (x)

Find the Laplace transform os sin t cos t.

Discuss the simple application of Fourier transform.

What do you know about Green's function? Explain it.

Find the Green's function for the boundary value problem

d²y/dx² – k²y = f (x)

with boundary condition, y (±∞) = 0.

Obtain the polar form of cauchy Riemann equations.

Show that log z is analytic function except at z = 0.

State Bessel's differential equation. Find the solution of its equation.

What is orthogonal curvilinear coordinate ? Obtain an expression for the curl in terms of orthogonal curvilinear coordinates.

(a) Find the Fourier transform of f(x) if

f(x) = { e^-ax a < x < b
0 x < a, x > b }

(b) Find the Laplace transform of the function F(t) = (e^at - 1) / a

Find L{t cos at}

Explain Green's function for poission's equation and find solution of poission's equation?

Discuss Green's function for Quantum Mechanical Scattering Problem.

State and Prove Cauchy integral formula.

State and prove cauchy Residue theorem.