Using the recurrence relation, show that

Determine the Fourier transform of the normalized Gaussian distribution

Obtain the eigenvalues and the corresponding eigenvectors of the matrix:

Using appropriate figures, show that the group S₃ is not commutative.

Using recurrence relation, show that

where J_n(x) is the Bessel function of the first kind. 

i) Show that the series  converges for |z|< 1 and find its sum.

ii) Obtain the analytic function whose real part u(x, y ) e^x cos y.

Using Cauchy’s residue theorem, evaluate the integral