1000 Solved Problems in Modern Physics

24 1 Mathematical Physics

1.25 Use the Beta functions to evaluate the definite integral

∫π/ 2

0 (cosθ)

rdθ

1.26 Show that:

(a)Γ(n)Γ(1−n)=sin(πnπ);0<n< 1

(b)|Γ(in)|^2 =nsinπh(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 − 11

03 − 1

00 2

⎞

⎠

1.29 Given below the set of matrices:

A=

(

− 10

0 − 1

)

,B=

(

01

10

)

,C=

(

20

02

)

,D=

(√

3

2

1

2

−^12

√

3

2

)

what is the effect when( A,B,CandDact separately on the position vector

x

y

)

?

1.30 Find the eigen values of the matrix:
⎛
⎝

6 − 22

− 23 − 1

2 − 13

⎞

⎠

1.31 Diagonalize the matrix given in Problem 1.30 and find the trace (Tr=λ 1 +
λ 2 +λ 3 )

1.32 In the Eigen vector equation AX=λX, the operator A is given by

A=

[

32

41

]

Find:

(a) The Eigen valuesλ

(b) The Eigen vectorX

(c) The modal matrixCand it’s inverseC−^1

(d) The productC−^1 AC

1.2.5 MaximaandMinima

1.33 Solve the equationx^3 − 3 x+ 3 =0, by Newton’s method.

1.34 (a) Find the turning points of the functionf(x)=x^2 e−x
2
.
(b) Is the above function odd or even or neither?