Loyola College Supplementary Physics April 2006 Mathematical Physics Question Paper PDF Download

 

 

             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

SUPPLEMENTARY SEMESTER EXAMINATION – JUN 2006

M.Sc. DEGREE EXAMINATION

                                            PH 2803/PH 2900 – MATHEMATICAL PHYSICS

 

 

 

Date & Time : 27/062006/9.00 – 12.00         Dept. No.                                                       Max. : 100 Marks

 

                                                                PART – A                                       (10´ 2=20 marks)

      Answer ALL questions.

 

  1. Starting from the general equation of a circle in the xy plane, A(x2 +y2) + Bx + Cy +D=0 arrive at the zz* representation for a circle.
  2. State Cauchy’s integral formula for derivatives
  3. Develop Taylor’s series of about z = -1.
  4. Express in the form of a+ib
  5. Show that the Dirac delta function .
  6. State convolution theorem.
  7. Solve the differential equation ’ + .
  8. Obtain the orthonormalising constant for the series in the interval     (-L, L).
  9. Evaluate using the knowledge of Gamma function.
  10. Generate L3 (x) and L4(x) using Rodrigue’s formula for Laugerre

 

 

 

                                                                PART – B                                      (4´ 7.5=30 marks)

      Answer any FOUR.

 

  1. Obtain Cauchy Rieman equations from first principles of calculus of complex numbers.
  2. State and prove Cauchy’s residue theorem
  3. Develop half-range Fourier sine series for the function f (x) = x ; 0 < x < 2. Use the results to develop the series .
  4. Verify that the system y11 + ; y1(0) = 0 and y (1) = 0 is a Sturm-Liouville System. Find the eigen values and eigen functions of the system and hence form a orthnormal set of functions.
  5. (a) If f (x) = obtain Parseval’s Identity
    where  Pk (x) stands for Legendre polynomials.
  • Prove that  (x) = 2n – 1 Hn (x) where Hn (x) stands for Hermite polynomials.(4+3.5)

 

 

                                                               PART – C                                      (4´12.5=50 marks)

Answer any FOUR.

 

  1. Show that u (x, y) = Sin x Cosh y + 2 Cos x Sinhy + x2 +4 xy – y2 is harmonic Construct f (z) such that u  + iv is analytic.
  2. (a)  Evaluate  using contour integration.

(b)  Using suitable theorems evaluate  c : .             (7+5.5)

  1. (a) The current i and the charge q in a series circuit containing an inductance L and
    capacitance c and emf E satisfy the equations L  and i = . Using
    Laplace Transforms solve the equation and express i interms of circuit parameters.
  • Find , where L-1 stands for inverse Laplace transform.                 (3.5)
  1. Solve the boundary value problem . with Y (0, t) = 0; yx (L, t) = 0
    y (x, 0) = f (x) ;  yt (x, 0)  = 0  and
  2. Solve Bessels differential equation using Froebenius power series method.

 

 

 

 

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Loyola College M.Sc. Physics April 2003 Mathematical Physics Question Paper PDF Download

 

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

M.Sc. DEGREE EXAMINATION – PHYSICS

SECOND SEMESTER – APRIL 2003

PH 2803 / PH 825  –  MATHEMATICAL PHYSICS

 

28.04.2003

1.00 – 4.00                                                                                                      Max : 100 Marks

                                                                PART – A                                       (10´ 2=20 marks)

      Answer ALL questions.

 

  1. Starting from the general equation of a circle in the xy plane A (x2 +y2) + Bx + Cy +D=0 arrive at the z z* representation for a circle.
  2. State Liouville’s theorem.
  3. Develop Laurent series of about z = -2.
  4. Write the Jacobian of the transformation .
  5. Show that the Dirac delta function .
  6. State convolution theorem.
  7. Solve the differential equation + .
  8. Obtain the orthonormalising constant for the series in the interval     (-L, L).
  9. Evaluate using the knowledge of Gamma function.
  10. Generate L2 (x) and L3 (x) using Rodrigue’s formula for laugerre

 

 

 

                                                                PART – B                                      (4´ 7.5=30 marks)

      Answer any FOUR.

 

  1. Obtain Cauchy Rieman equations from first principles of calculus of complex numbers.
  2. Determine a function which maps the indicated region of w plane on to the upper half of the z – plane

v                                                                           y

w plane                                                                    z plane

p                              T

 

Q                      s         u                     p1            Q1                 S1       T1         X

-b                     +b                                                 -1                 +1

  1. Develop half-range Fourier sine series for the function f (x) = x ; 0 < x < 2. Use the results to develop the series .
  2. Verify that the system y11 + ; y1(0) = 0 and y (1) = 0 is a Sturm-Liouville System. Find the eigen values and eigen functions of the system and hence form a orthnormal set of functions.
  3. (a) If f (x) = obtain Parseval’s Identity
    where  Pk (x) stands for Legendre polynomials.
  • Prove that  (x) = 2n – 1 Hn (x) where Hn (x) stands for Hermite polynomials.(4+3.5)

 

 

                                                               PART – C                                      (4´12.5=50 marks)

Answer any FOUR.

 

  1. Show that u (x, y) = Sin x Coshy + 2 Cos x Sinhy + x2 +4 xy – y2 is harmonic Construct f (z) such that u  + iv is analytic.
  2. (a)  Evaluate  using contour integration.

(b)  Using suitable theorems evaluate  c : .                                  (7+5.5)

  1. (a) The current i and the charge q in a series circuit containing an inductance L and
    capacitance C and emf E satisfy the equations L  and i = . Using
    Laplace Transforms solve the equation and express i interms of circuit parameters.
  • Find , where L-1 stands for inverse Laplace transform.                 (3.5)
  1. Solve the boundary value problem . with Y (0, t) = 0; yx (L, t) = 0
    y (x, 0) = f (x) ;  yt (x, 0)  = 0  and  and Interpret physically.
  2. Solve Bessels differential equation using Froebenius power series method.

 

 

 

 

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Loyola College M.Sc. Physics Nov 2003 Mathematical Physics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M. Sc. DEGREE EXAMINATION – PHYSICS

SECOND SEMESTER – NOVEMBER 2003

PH 2803 / PH 825 – MATHEMATICAL PHYSICS

 

15.11.2003                                                                                                  Max.   : 100 Marks

1.00 – 4.00

 

PART – A

 

Answer ALL the questions.                                                                              (10 x 2 = 20)

 

  1. Express x2 + y2 = 25 in zz* and reirepresentation.

 

  1. State Liouville’s theorem.
  2. Find Laurent Series of at z = 1 and name the Singularity.
  3. Find the Jacobian of transformation of w = z2.

 

  1. Find L (FÎ (t)), where FÎ (t) represents Dirac delta function.

 

  1. State parseval’s theorem.

 

  1. Obtain the orthonormalizing constant for the set of functions given by ; n = 1, 2, 3  . . . .   in the interval –L to +L.

 

  1. Solve the differential equation y¢ + k l2 y = 0.

 

  1. Write Laplace equation in spherical polar co-ordinates.

 

  1. Using Rodrigue’s formula for Legendre polynomials, evaluate P3(x).

 

PART – B

 

Answer any FOUR.                                                                                         (4 x 7.5 = 30)

 

  1. Derive the necessary conditions for a function to be analytic.

 

  1. Find the residues of f(z) = at its poles.

 

  1. Expand f(x) = sin x, 0 < x < p in a fourier cosine series and hence prove that

 

 

(P.T.O)

-2-

 

 

 

 

  1. Verify that the system y¢¢ + ly = 0; y¢ (0) = 0 and y(1) = 0 is a Sturm-Liouville system. Find the eigenvalues and eigenfunctions of the system. Prove that eigenfunctions are orthogonal.

 

  1. Prove that Ln+1(x) = (2n + 1 – x)Ln(x) – n2  Ln-1(x) where L’s stand for Laugerre polynomials.

 

PART – C

 

Answer any FOUR.                                                                                       (4 x 12.5 = 50)

 

  1. (i) Evaluate along

 

  1. the parabola x = 2t, y = t2 + 3

 

  1. straight lines from (0, 3) to (2,3) and then from (2,3) to (2,4) and

 

  1. a straight line from (0, 3) to (2, 4).    (7.5)

 

(ii) State and prove Poisson’s Integral formula for a circle.                                       (5)

 

  1. Using contour Integration, evaluate for a>|b|.

 

  1. An Inductor of 2 henrys, a resistor of 16 ohms and a capacitor of 0.02 farads are connected in series with an e.m.f E volts. Find the charge and current at any time t>0 if a) E =  300 V and   b) E = 100 sin 3t Volts

 

  1. Generate Set of orthonormal functions from the sequence 1, x, x2, x3 . . . . using Gram-Schmidt orthonormalization process.

 

  1. Write Bessel’s differential equation and obtain the standard solution.

 

 

 

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Loyola College M.Sc. Physics April 2008 Mathematical Physics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

FG 31

M.Sc. DEGREE EXAMINATION – PHYSICS

SECOND SEMESTER – APRIL 2008

    PH 2900 / 2803 – MATHEMATICAL PHYSICS

 

 

 

Date : 26/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

Answer ALL questions.                                                        (10 x 2 = 20 marks)

 

  1. State Cauchy’s integral formula for derivative.
  2. Check whether is analytic or not.
  3. Find the Taylor expansion for a function
  1. State Laurent’s Theorem.
  2. Give two examples of periodic functions.

Is periodic or not?

  1. Find the Fourier transform of
  2. Solve , where is a constant.
  3. Are the set of functions are orthogonal in the limit . Expalin why?
  4. Show that .
  5. Using the knowledge of Gamma & Beta functions evaluate,

 

PART – B

Answer any FOUR questions.                                             (4 x 7.5 = 30 marks)

 

  1. Derive Cauchy-Riemann equation in polar Co-ordinate System.
  2. a) Using Cauchy’s Integral formula,

evaluate .

  1. b) If is analytic in a closed region, show that .
  1. Expand      in a series of sines with a period of 8.
  2. Show that the eigen functions belonging to two different eigen values are orthogonal with respect to R(x) in (a, b)
  3. Prove that

 

 

PART – C

Answer any FOUR questions.                                             (4 x 12.5 = 50 marks)

 

  1. Derive Poisson’s Integral formula for circle.
  2. Using Cauchy’s Residue theorem,

evaluate

  1. a) Find the Fourier coefficients corresponding to the function

with the Period 10.

  1. b) Prove that
  1. Solve the boundary value problem,
  1. Using the method of Frobenius, solve Laugerre’s differential equation.

 

 

 

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Loyola College M.Sc. Physics April 2012 Mathematical Physics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – PHYSICS

SECOND SEMESTER – APRIL 2012

PH 2812 – MATHEMATICAL PHYSICS

 

 

Date : 21-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer ALL questions:                                                                                                              (10×2=20)

  • Evaluate the complex line integral around the closed loop C: |z| = 1.
  • Determine the residue at Z= 0 and at Z = I of the complex function f (z) = .
  • Define Dirac delta function . What is its Laplace transform?
  • Express the function f (t) = 2 if 0<t<π, f(t) = 0 if π<t<2π and f(t) = sin t if t >2π in terms of the unit step function.
  • What are the two possible initial conditions in the vibration of a rectangular membrane? Explain the symbols used
  • Solve .
  • Use the Rodrigue’s formula to evaluate the 3rd degree Legendre polynomial .
  • State the orthonormality property of the Hermite polynomials.
  • List the four properties that are required by a group multiplication.
  • What is irreducible representation of a group?

 

PART – B

Answer any FOUR questions:                                                                                                 (4×7.5 = 30)

 

  • Verify the Cauchy’s integral theorem for the contour integral for the contour C: the triangle with vertices at 0, 1, and 1+i.
  • A capacitor of capacitance C is charged so that its potential is V0. At t = 0 the switch in figure is closed and the capacitor starts to discharge through the resistor of resistance R. using the Laplace transformation, find the charge q(t) on the capacitor.
  • Use the method of separation of variables to solve the partial differential equation , where u( x,0) = 6 e-3x.
  • (a) Prove that J-n(x) = (-1)n Jn(x) if n is a positive integer where Jn(x) is the Bessel function of first kind.

(b) Determine the value of J -1/2(x).                      ( 4 ½ +3)

  • Work out the multiplication table of the symmetry group of the proper covering operations of a square. Write down all the subgroups and divide the group elements into classes. What are the allowed dimensionality of the representation matrices of the group?

 

PART – C

Answer any FOUR questions:                                                                                                  (4×12.5 =50)

  • (a) Using the contour integration, evaluate the real integral,

(b) Evaluate the following integral using Cauchy’s integral formula dz, where C is the circle |Z |= 3/2.                   ( 6 ½ + 6)

  • Find the Fourier transform of (i) f(x) = exp( -x2) and (ii) f(x) =1 – |x| if |x| <1 and f(x) = 0 for |x| >1

( 6 ½ + 6)

  • Solve the one- dimensional wave equation by the separation of variable technique and the use of Fourier series. The boundary conditions are u(0,t) =0 and u(L,t) = 0 for all t and the initial conditions are u ( x,0) = f(x) and ∂u/∂t = g(x) at t =0. ( Assume that u (x,t) to represent the deflection of stretched string and the string is fixed at the ends x = 0 and x = L)
  • (a) Solve the Legendre differential equation (1 – x2) – 2x  + n (n+1)y = 0 by the power series method.

(b) Establish the orthonormality relation where  is the  Legendre polynomial of order n.                 ( 6 ½ + 6)

  • (a) Prove that any representation by matrices with non-vanishing determinants is equivalent to

a representation by unitary matrices.

(b)  Enumerate and explain the symmetry elements of CO2, H2O and NH3 molecules. ( 6 ½ +6)

 

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Loyola College M.Sc. Physics Nov 2016 Mathematical Physics Question Paper PDF Download

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Loyola College B.Sc. Physics April 2008 Mathematical Physics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

FG 17

 

FOURTH SEMESTER – APRIL 2008

PH 4502 – MATHEMATICAL PHYSICS

 

 

 

Date : 26/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART-A

Answer ALL questions                                                                     (10×2=20 marks)

 

  1. What is the principal value of the complex number z=1+i?
  2. Write down the equation of the circle in the complex plane centered at ‘a’ with radius ‘r’.
  3. Evaluate .
  4. What is a single valued function in a complex region.
  5. Find ‘c’ if is solution to the equation
  1. Write down a homogeneous first order partial differential equation.
  2. Define the Fourier sine transform of a function f(x).
  3. If is the Fourier transform of f(x), what is the Fourier transform of .
  4. Define the shift operator on f(x) by ‘h’.
  5. Write down the Simpson’s 1/3 rule for integration.

 

 

PART-B

Answer any FOUR questions                                                          (4×71/2=30 marks)

 

  1. Determine the roots of  and  and locate it in the complex plane.
  2. If ‘C’ is a line segment from -1-i to 1+i, evaluate .
  3. Derive the partial differential equation satisfied by a vibrating elastic string subject to a     tension ‘T’.
  4. Obtain the Lagrange’s interpolation formula for following table:
  1. Find the Fourier sine transform of exp(-at).

 

 

PART-C

Answer any FOUR questions                                                          (4×121/2=50 marks)

 

  1. a) Derive the Cauchy Riemann equation for a function to be analytic.          (5m)
  1. b) Show that the function is harmonic and hence

construct the corresponding analytic function.                                               (71/2m)

 

 

 

  1. a) State and prove Cauchy’s integral theorem.                                               (5m)
  1. b) Verify the Cauchy’s integral theorem for the integral of taken over the boundary of the rectangle with vertices -1, 1, 1+i and -1 +i in the counter clockwise sense. (71/2m)
  1. Solve the heat equation , subject to the conditions u(x=0,t)=0 and u(x=L,t)=0       for all ‘t’.

 

  1. a) State and prove the convolution theorem for Fourier Transforms.             (2+3=5m)
  1. b) Find the Fourier transform of the function f(x) defined in the interval –L to +L, as

(71/2m)

 

  1. Given the following population data, use Newton’s interpolation formula to find the population for the years 1915 and 1929

 

(Year, Population (in Thousands)): (1911, 12) (1921, 15) (1931, 20), (1941, 28).

 

 

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Loyola College B.Sc. Physics April 2008 Mathematical Physics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

FG 24

 

SIXTH SEMESTER – APRIL 2008

PH 6604 / 6601 – MATHEMATICAL PHYSICS

 

 

 

Date : 21/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL the questions:                                                                          (10 x 2 = 20)

  1. Plot  for a fixed ‘r’ and .
  2. Find  a)     b) .
  3. Give the condition on a function f(z)=u(x,y) + i v(x,y) to be analytic.
  4. Evaluate along a straight line from i to 1+i.
  5. Write down the Lplace’s equation in two dimensions in polar coordinates.
  6. Write down the equation for one dimensional heat flow.
  7. State Parsavel’s theorem.
  8. Define Fourier cosine transform.
  9. Give trapezoidal formula for integration.
  10. Define the forward and backward difference operators.

PART – B

Answer any FOUR  questions.                                                                    (4 x 7\ = 30)

  1. Show that the following function  is harmonic and hence find the corresponding analytic function, .
  2. Prove Cauchy’s integral theorem.
  3. Find D’ Alembert’s solution of the vibrating string.
  4. State and prove convolution theorem in Fourier transform.   (2+5\=7\)
  5. Use Simpson’s 1/3 rule to find correct to two decimal places, taking step size h=0.25.

PART – C

Answer any FOUR   questions.                                                                   (14 x 12\ = 50)

  1. a) Determine  and plot its graph                                                           (3\)
  1. b) Perform the following operations i) ii) and locate these values in the complex plane. (4+5=9)
  1. a) State and prove Cauchy’s integral formula.
  1. b) Integrate in the counter clockwise sense around a circle of radius 1 with centre at z=1/2. (2+5+5\=12\)
  1. Derive the Laplace’s equation in two dimensions and obtain its solutions.
  2. Find the Fourier transform and the Fourier cosine transform of the function

.

  1. Estimate the value of f(22) and f(42) from the following data by Newton’s interpolation:
x 20 25 30 35 40 45
f(x) 354 332 291 260 231 204

 

 

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Loyola College B.Sc. Physics April 2009 Mathematical Physics Question Paper PDF Download

       LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

XC 22

SIXTH SEMESTER – April 2009

PH 6604 – MATHEMATICAL PHYSICS

 

 

 

Date & Time: 21/04/2009 / 9:00 – 12:00     Dept. No.                                                       Max. : 100 Marks

 

 

PART-A                                                             (10 X 2 =20 MARKS)

 

ANSWER ALL QUESTIONS

 

1). Write down the triangle inequality for two complex numbers z1 and z2.

2). Write down the complex representation for a circle of radius 2 units.

3). State Cauchy Riemann conditions for a function f(x,y) = u(x,y) + i v(x,y) to be analytic.

4). Define simply connected and multiply connected domains in a complex plane.

5). If u1 and u2 are two solutions of a homogeneous differential equation what can you say about

u = a u1 + b u2,   with `a’ and ‘b’  constants.

6). Write down the two dimensional wave equation for a wave with velocity 1 m/s.

7). In the expansion for f(x) = a0 + n cos (nx) + n sin (nix) , write a0  in terms of f(x).

8). If f(x) is an even function of period 2, what happens to the Fourier sine coefficients.

9). Write down the Trapezoidal rule for integration of a function f(x) between x0 and x0 +h.

10.Write down the relationship between the shift operator `E’ and the forward difference

operator.

 

PART-B                                                             (4 X 7.5 = 3O MARKS)

 

ANSWER ANY FOUR QUESTIONS.

 

11). If f(z) = 3z2 + z, evaluate f(z) for a). z = 2 + i and z = -4 + 2 i and locate these points in the

complex plane.

12). State and prove Cauchy’s integral theorem.

13). Obtain the Laplace equation in two dimensions in terms of the polar coordinates.

14). If F(s) is the Fourier transform of f(x) find the Fourier transform of f(ax) and f(x-u), with

`a’ and `u’ being constants.

15). Using Euler method, solve the following differential equation to find y(0.4), given ,

with y(0)=1 and h = 0.1. Compare your result with the exact solution.

 

PART-C                                                             (4 x 12.5 = 50 MARKS)

ANSWER ANY FOUR QUESTIONS.

 

  1. What do you mean by conjugate harmonic functions? If the following functions are

harmonic, find their conjugate functions, f(x,y) = u(x,y) + i v(x,y),

a). u (x,y) = e x cos (y) ; b). v = xy.

17). Evaluate the following integrals over the  unit circle.

  1. a) and  b).

18). (i). Find `a’ and ‘b’ if u(x,y) = a x2 – b y2 is solution of the Laplace equation in two

dimensions.

(ii). Derive the partial differential equation for small transverse displacement `u’ of an

elastic string.

19). (i). State and prove Parseval’s identities for Fourier transforms.

(ii). Find the Fourier transform of f(x) = 2  for –a < x < a and f(x) = 0 for all other values.

20). From the following census data find the population for the year 1895 and 1906

 

Year 1891 1901 1911 1921 1931
Population

(in thousands)

46 66 81 93 101

 

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Loyola College B.Sc. Physics April 2009 Mathematical Physics Question Paper PDF Download

       LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

XC 15

B.Sc. DEGREE EXAMINATION – PHYSICS

FOURTH SEMESTER – April 2009

PH 4502 – MATHEMATICAL PHYSICS

 

 

 

Date & Time: 24/04/2009 / 9:00 – 12:00  Dept. No.                                                  Max. : 100 Marks

 

 

 

PART-A                                                             (10 x 2 = 20 MARKS)

 

ANSWER ALL QUESTIONS.

 

1). Given z1 = 2 – i and z2 = 2 + i find z1* z2 .

2). Check if the function f(z) = x + i y  is analytic.

3). Evaluate .

4). Explain the property of linearity in complex line integral.

5). Define the eigen value problem for the operator .

6). Write down the two dimensional wave equation.

7). Give the Parseval’s identity for Fourier transforms.

8). Define Fourier sine transform.

9). Why is Lagrange’s interpolation advantageous over Newton’s interpolation?

10). Write down Simpson’s 1/3 rule for integration.

 

PART-B                                                             ( 4 x 7.5 = 30 MARKS).

 

ANSWER ANY FOUR QUESTIONS.

 

11). Simplify the following a). (4+ 2i) (2 + i) ; b). 4[(2+2i)/(2-2i)]2 – 3[(2-2i)/(2+2i)]2,

Locate these points in the complex plane.

12). Verify Cauchy’s integral theorem for the integral of  z 2 over the boundary of the

rectangle with vertices (0,0) , (1,0) , (1,1), (0,1) in the counterclockwise sense.

13). Find D’Alembert’s solution of the wave equation for a vibrating string.

14). Prove the following for the Fourier transforms F{f(ax)}= (1/a)F(s/a) and F{f’(x)}= is

F(s), here F(s) is the Fourier transform of f(x) and the prime denotes differentiation

with respect to `x’.

15). Use Euler method to solve with y(0) = 2 Find y(0.2) with h = 0.1.

 

PART-C                                                             (4 x 12.5 = 50 MARKS)

 

ANSWER ANY FOUR QUESTIONS.

 

16). State and prove Cauchy’s integral formula.

17). Derive the Cauchy Riemann equation for a complex function to be analytic. Express

it in polar coordinates.

18). Explain the method of separation of variables to solve the one dimensional wave

equation .

Check whether u = x2 – y2 satisfies the two dimensional Laplace equation.

19). (a). State and prove the convolution theorem for the Fourier transforms.

(b). Find the Fourier sine transform of e-ax.

  1. (a). Given y = sin (x ) , generate the table for x = 0 /4 and /2 Find the value of

sin (/6). using Lagrange’s interpolation method.

(b). For the given data calculate the Newton’s forward difference table.

(x,y): (0,0), (1,2), (2,6), (3,16).

 

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Loyola College B.Sc. Physics April 2011 Mathematical Physics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

FOURTH SEMESTER – APRIL 2011

PH 4504/PH 4502/PH 6604 – MATHEMATICAL PHYSICS

 

 

Date : 07-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART-A

Answer ALL questions.                                                                                             (10 x 2 = 20 marks)

 

  1. Given z1 = a – i and z2 = a + i fine z1* z2, for any real ‘a’.
  2. Verify that f(z) = z is analytic.
  3. State two conditions for a function to be Fourier transformed.
  4. Define the eigen value problem for the operator
  5. Express the Laplacian in polar coordinates.
  6. State Cauchy’s integral theorem.
  7. Evaluate , ‘c’ is circle of radius 1.
  8. State Parseval’s theorem.
  9. Write down the difference operator and the shift operator.
  10. Write down trapezoidal rule for integration.

 

PART-B

 

Answer any FOUR questions.                                                                                   (4 x 7.5  = 30 marks)

 

  1. a). Show that |z|2 = 1 describes a circle centered at the origin with radius 1.

b). Simplify (1+i)(2+i) and locate it in the complex plane.

  1. Verify the Cauchy’s integral theorem for along the boundary of a rectangle with vertices

(0,0) , (1,0), (1,1) and (0,1) in counter clock sense.

  1. Find DAlembert’s solution of the wave equation for a vibrating string.
  2. If f(s) is the Fourier transform of f(x), show that F{f(ax)} = (1/a)F(s/a) and

F{f’(x)} = is F(s). Here the prime denotes differentiation with respect to ‘x’.

  1. Use Euler’s method to solve, given y(0) = 1, find y(0.04) with h = 0.01.

PART-C

Answer any FOUR questions.                                                                                   (4 x 12.5 = 50 marks)

 

  1. a) Establish that the real and complex part of an analytic function satisfies the Laplace equation.
  2. b) Prove that is harmonic and find its conjugate function.                                             (6+6.5)
  3. Verify

a). for f(z) = z, with z0 = -1-i and z= 1+i.

b).

for f(z) = 3z and g(z) = -3,  and any real constants c1 and c2.

  1. Using the method of separation of variables obtain the solution for one dimensional

heat equation. , with u(l,t) = 0 and u(0,t)=0.

  1. a) State and prove convolution theorem for Fourier transforms.
  2. b) Find the Fourier sine transform of .
  3. Derive the Newton’s forward interpolation formula and deduce the Trapezoidal and Simpson’s rule

for integration.

 

 

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Loyola College B.Sc. Physics April 2012 Mathematical Physics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

FOURTH SEMESTER – APRIL 2012

PH 4504/4502/6604 – MATHEMATICAL PHYSICS

 

 

 

Date : 21-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART-A

 

Answer ALL questions:                                                                                           (10 x 2 = 20 marks)

 

  1. Given z1 = a – i and z2 = a + i find z1* z2, for any real ‘a’.
  2. Verify that f(z) = x2-y2+2ixy is analytic.
  3. Evaluate.
  4. Define the eigen vale problem for the operator.
  5. Find ‘c’ if u(x,t) = x2+at2 is a solution to the wave equation

.

  1. What is singular point of a complex function in a region.
  2. Write down a homogeneous first order partial differential equation in two variables.
  3. State Parsavel’s theorem.
  4. Write down the difference operator for f(x) by ‘h’.
  5. Write down trapezoidal rule for integration.

PART-B

 

Answer any FOUR questions:                                                                                (4 x 7.5 = 30 marks)

 

  1. a). Show that |z-i|2 = 1 describes a circle centered at the (0,i) with radius 1.

b). Simplify (1+i)(2+i) and locate it in the complex plane.

  1. If ‘C’ is a line segment from -1-i to 1+i evaluate .
  2. Derive the partial differential equation satisfied by a vibrating elastic string subject to

a tension ‘T`.

  1. If F(s) is the Fourier transform of f(x), show that F{f(ax)} = (1/a)F(s/a) and

F{f’(x)} = is F(s). Here the prime denotes differentiation with respect to ‘x’.

  1. Obtain the Lagrange’s interpolation polynomial of degree two for the following data:

(x,y): (0,0),(1,3),(2,9)

 

 

 

PART-C

 

Answer any FOUR questions:                                                                                 (4 x 12.5 = 50 marks)

 

  1. Establish that the real and complex part of an analytic function satisfies the Laplace equation.

 

  1. a) State and prove Cauchy’s integral theorem.
  2. b) Verify the integral theorem for , where c is a circle of radius 1.
  3. Obtain the Laplacian operator in polar form from the Cartesian form.
  4. a) State and prove convolution theorem for the Fourier transforms.
  5. b) Find the Fourier sine transform of .
  6. Derive the Newton’s forward interpolation formula and deduce the Trapezoidal and Simpson’s rule

for integration.

 

 

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Loyola College B.Sc. Physics Nov 2012 Mathematical Physics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

FOURTH SEMESTER – NOVEMBER 2012

PH 4504/4502/6604 – MATHEMATICAL PHYSICS

 

 

 

Date : 03/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART-A

 

Answer ALL questions:                                                                                           (10 x 2 = 20 marks)

 

  1. If z* = – z, what can you conclude about ‘z’?
  2. What is the geometrical meaning of the curve |z| = r.
  3. Show =2πi. Where ‘c’ is a circle of unit radius, with centre at (0, 0)
  4. Define the eigenvalue problem for the operator.
  5. Show u = x + ct satisfies the equation .
  6. What is singular point of a complex function in a region?
  7. Write down a homogeneous first order partial differential equation in two variables.
  8. Define Fourier sine transform of the function f(x).
  9. Write down the backward difference operator for f(x) by ‘h’.
  10. Write down Simpson’s 1/3 rd rule for integration.

 

PART – B

 

Answer any FOUR questions:                                                                                 (4 x 7.5 = 30 marks)

 

  1. a). Plot the function x + i y for (x,y) varying in the region (0,1).

b). Simplify (1+i)(2+i) and locate it in the complex plane.

  1. If ‘c’ is a line segment from -i to +i , evaluate .
  2. Discuss the D’Alembert solution of the wave equation.
  3. If f(s) is the Fourier transform of f(x), show that F{f(ax)} = (1/a)F(s/a) and

F(s). Here the prime denotes differentiation with respect to ‘x’.

  1. Deduce a second order polynomial using Newton interpolation formula for:

(x,y): (0,0),(1,3),(2,9).

PART-C

Answer any FOUR questions:                                                                                 (4 x 12.5 = 50 marks)

 

  1. Deduce the Cauchy Riemann conditions in polar coordinates for complex function, to be analytic and

establish that analytic function satisfy Laplace’s equation.

  1. a). State and prove Cauchy’s integral formula

b). Verify the Cauchy’s integral theorem for , where c is a circle of radius 1.

  1. Discuss the solution of the two dimensional Laplace equation.
  2. a). State and prove convolution theorem for the Fourier transforms.

b). Find the Fourier sine transform of .

  1. For the following data evaluate by (i) Trapezoidal rule,

(ii) Simpson’s 1/3 rd rule.

 

(x, f(x)): (1, 2.105) (2, 2.808) (3, 3.614) (4, 4.604) (5, 5.857) (6, 7.451) (7, 9.467).

 

 

 

 

 

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