24
1 Mathematical Physics
1.25 Use the Beta functions to evaluate the definite integral
π/2
0 (cos θ )
r dθ
1.26 Show that:
(a) Γ(n)Γ(1 − n) =
π
sin(nπ )
; 0 < n < 1
(b) |Γ(in)|
2
=
π
n sin h(nπ)
1.2.4 Matrix Algebra
1.27 Prove that the characteristic roots of a Hermitian matrix are real.
1.28 Find the characteristic equation and the Eigen values of the matrix:
⎛
⎝
1 −1 1
0 3 −1
0 0 2
⎞
⎠
1.29 Given below the set of matrices:
A =
−1 0
0 −1
, B =
0 1
1 0
, C =
2 0
0 2
, D =
√
3
2
1
2
−
1
2
√
3
2
what is the effect when A, B, C and D act separately on the position vector
x
y
?
1.30 Find the eigen values of the matrix:
⎛
⎝
6 −2 2
−2 3 −1
2 −1 3
⎞
⎠
1.31 Diagonalize the matrix given in Problem 1.30 and find the trace (T r = λ 1 +
λ 2 + λ 3 )
1.32 In the Eigen vector equation AX = λX, the operator A is given by
A =
3 2
4 1
.
Find:
(a) The Eigen values λ
(b) The Eigen vector X
(c) The modal matrix C and it’s inverse C
−1
(d) The product C
−1 AC
1.2.5 Maxima and Minima
1.33 Solve the equation x
3
− 3x + 3 = 0, by Newton’s method.
1.34 (a) Find the turning points of the function f (x) = x
2 e
−x
2 .
(b) Is the above function odd or even or neither?