Mathematical Physics Important Questions

Bsc/Msc— Physics/Maths 

Unit 1: Differential Equations 

Bessel function 

1. Solution of bessel differential equation.
2. Bessel function of the first and second kinds of order n.
3. Recurrence relation for Jn(x).
4. Orthogonality property of bessel's functions. 
5. Generating function for Jn(x).
6. Bessel's integral.

Legendre's function 

1. Solution of Legendre's differential equation.
2. Legendre functions of the first and second kinds of order n and their properties.
3. Recurrence relations for Pn(x).
4. Orthogonality property of Legendre function. 
5. Generating function for Pn(x) legendre's polynomial.
6. Legendre polynomial integral.
7. Rodrigue's formula for Pn(x) (Legendre polynomials).
8. Associate Legendre's function.

Hermite's function 

1. Solution of Hermite's differential equation.
2. Hermite functions of the first and second kinds of order n.
3. Recurrence relations for Hn(x).
4. Orthogonality property of Hermite functions. 
5. Generating function for Hn(x) (Hermite's polynomials).
6. Rodrigue's formula for Hn(x) (Hermite polynomials).
7. Hermite's integral.

Laguerre's function

1. Solution of laguerre's differential equation.
2. Laguerre polynomials of the first and second kinds of order n.
3. Recurrence relations for Ln(x).
4. Orthogonality property of laguerre functions. 
5. Generating function for Ln(x).
6. Laguerre integral.

Curvilinear co-ordinate System 

1. Specific cases of Cartesian co-ordinate system 
2. Cylindrical co-ordinate system 
3. Spherical co-ordinate system 

Unit 2: Integral Transform

1. Explain Fourier sine and cosine transform. Discuss the Fourier transform of derivatives.
2. Find the Fourier transform of F(x) defined as F(x)= any condition.
3. State and prove Convolution theorem of Fourier transform.
4. Explain Laplace transform and shows that  "any equation value'
5. Define Laplace transform establish the relation between Fourier transform and Laplace transform.
6. Disscus about an Application of dumped harmonic oscillator. 

Unit 3: Green's Function 

1. Find the Green's Function for one dimensional problem.
2. What do you understand by Green's Function? 
3. Disscus the Green's Function in electrostatic  boundary value problem.
4.  Discuss the Green's Function for quantum mechanical scattering problem.
5. Explain Green's Function symmetry properties.
6. Explain the Green's Function method of solving boundary value problems. And explain its symmetry properties.
7. Explain eigen function expansion of Green's Function.
8. Construct the Green's Function by Fourier transform method.

Unit 4: Complex Variables

1. Define Analytic Function . Derive Sufficient and Necessary conditions for a function to be analytic.
2. State and prove taylors theorem.
3. State and prove Cauchy's Theorem.
4. Explain Taylor series expansion of a function F(z) with centre at z⁰.
5. State and prove Cauchy's Residue Theorem and find the residue of "a equation".
6. State and Derive Cauchy's Integral formula.
7. Find contour integral of "a equation".
8. Explain jorden's lemma Integral.
9. Explain Maclaurin and laurent series and mapping.