Aligarh Muslim University (AMU) 2011 B.Sc Physics Mathematical Methods - Question Paper
4482
2010-2011 B.Sc. (HONS.) (PART - III) EXAMINATION (PHYSICS)
MATHEMATICAL METHODS
(PH-311)
Maximum Marks : 40 Duration : Three Hours
Note: Answer all questions.
1.(a) Find the analytic function f(z) if u(x, y) = x3 - Sxy2. . . [214]
(b) Evaluate & Zt*Z, the curve C is given by | z | = 2.
I z +4z + 3
(c) State and prove Cauchys integral theorem. [2]
OR
\ F21
l'.(a) Show that ck dz = 2ni f(z0) , where z0 is a point in the interior region
c z~zo
bounded by C and f(z) is analytic on C and within the region surrounded by C.
z [2]
(b) Find the residues of the function z-at each pole.
w z2 +4
(c) Evaluate the real integral [3]
2f de ll,
- e <1
- 1+ e cos 0 using calculus of residues.
as
2.(a) Show that the gradient of a scalar function <|> (x, y, z) is identified A a vector having [4] the direction of the maximum space rate of change of <(>.
(b) State and prove Gausss theorem. [2]
OR
2'.(a) Define curvilinear coordinates (qi, q2, q3) system. Show that square of the distance [4] ds between two neighbouring points in the orthogonal curvilinear coordinates can be
written as ds2 = (h; dq( f, where hjs are scale factors, i
(b) Show that [2]
V [r V (r)] = 3V(r)+r where V(r)is a central potential.
-2-
-rw
[4]
3.(a) Evaluate J e-*2 H(x)dx, where H(x) is Hermite polynomials.
(b) Show that Bessels functions Jn(x) and J_n(x) are linearly dependent for integer n. [3]
4-(a) Showthat yl _m(0, <(>)=(-1) y*+m(0,(|)). W
0>) For associated Legendre polynomials P (x), show that [4]
2q + l (q-m)l
5. Solve the wave equation for the vibration in a circular membrane of radius R, [7] clamped at the circumference, if its initial deflation u (r, 0) = f(r) and velocity
= g(r). Discuss the normal modes and sketch tht modes for m = 1 and m= 2
(r,t)
dtv '
t=0
6.(a) Solve the differential equation [2]
d2y(t) / \
using Laplace transform, if y(0) = 1 and y'(0) = 1.
(b) Solve the integral equation [4]
y(x) = l + A,J (l-3xz) y(z) dz.
o
OR
6'.(a) Classify the integral equations and give the example for each case. [3V4]
(b) Find the Fourier transform of e_a*2 (where a > 0).
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