1000 Solved Problems in Modern Physics

604 Appendix: Problem Index

1.2.3 Gamma and Beta Function

(a)Γ(Z+1)=ZΓ(Z)(b)Γ(n+1)=n! for positive integer 1.23

β(m,n)=Γ(m)Γ(n)/Γ(m+n)1.24

To evaluate a definite integral usingβfunctions 1.25

(a)Γ(n)Γ(1−n)=π/sin(nπ); 0<n< 1

(b)|Γ(in)|^2 =π/nsinh(nπ)

1.26

1.2.4 Matrix Algebra

Characteristic roots of Hermitian matrix are real 1.27

Characteristic equation and eigen values of the given matrix 1.28

Effect of a set of matrices on position vector 1.29

Eigen values of the given matrix 1.30

Diagnalization of a matrix and its trace 1.31

Eigen values, eigen vector, modal matrixCand its inverseC−^1 ,

productC−^1 AC

1.32

1.2.5 Maxima and Minima

Solution of a cubic equation by Newton’s method 1.33

(a) Turning points of f(x) (b) Whether f(x) is odd or even or neither 1.34

1.2.6 Series

Interval of convergence of series 1.35

Expansion of logxby Taylor’s series 1.36

Expansion of cosxinto infinite power series 1.37

Expansion of sin(a+x) by Taylor’s series 1.38

Sum of series 1+ 2 x+ 3 x^2 + 4 x^3 +...,|x|<11.39

1.2.7 Integration

(a)

∫

sin^3 xcos^6 xdx(b)

∫

∫ sin^4 xcos^2 xdx 1.40

(2x^2 − 3 x−2)−^1 dx 1.41

(a) Sketch of curver^2 =a^2 sin 2θ(b) area within the curve between

θ=0 andθ=π/ 2

1.42

∫

∫(x^3 +x^2 +2)(x^2 +2)−^2 dx 1.43

∞

0 4 a

(^3) (x (^2) + 4 a (^2) )− (^1) dx 1.44
(a)

∫

tan^6 xsec^4 xdx(b)

∫

tan^5 xsec^3 xdx 1.45

∫ 4

2 (2x+4)(x

(^2) − 4 x+8)− (^1) dx=ln 2+π 1.46
Area included between curvey^2 =x^3 and linex=41.47
Surface of revolution of hypocycloid aboutx-axis 1.48
∫a
0
∫√a (^2) −x 2
0 (x+y)dydx 1.49
Area enclosed between curvesy= 1 /xandy=− 1 /xand lines
x=1 andx= 2

1.50

∫

(x^2 − 18 x+34)−^1 dx 1.51

∫ 1

0 x

(^2) tan− (^1) xdx 1.52
(a) Area bounded by curvesy=x^2 +2 andy=x−1 and lines
x=−1 andx= 2
(b) Volume of solid of revolution obtained by rotating area enclosed
by linesx=0,y=0,x=2 and 2x+y=5 through 2πrad about
y-axis

1.53