1000 Solved Problems in Modern Physics

4 1 Mathematical Physics

Γ(n)=

∫∞

0

e−xxn−^1 dx (Re n>0) (1.11)

Γ(n+1)=nΓ(n) (1.12)

Ifnis a positive integer

Γ(n+1)=n! (1.13)

Γ

(

1

2

)

=

√

π;Γ

(

3

2

)

=

√

π

2

;Γ

(

5

2

)

=

3

4

√

π (1.14)

Γ

(

n+

1

2

)

=

1. 3. 5 ...(2n−1)

√

π

2 n

(n= 1 , 2 , 3 ,...) (1.15)

Γ

(

−n+

1

2

)

=

(−1)n 2 n

√

π

1. 3. 5 ...(2n−1)

(n= 1 , 2 , 3 ,...) (1.16)

Γ(n+1)=n!∼=

√

2 πnnne−n (Stirling’s formula) (1.17)

n→∞

Beta functionB(m,n) is defined as

B(m,n)=

Γ(m)Γ(n)

Γ(m+n)

(1.18)

B(m,n)=B(n,m) (1.19)

B(m,n)= 2

∫π/ 2

0

sin^2 m−^1 θcos^2 n−^1 θdθ (1.20)

B(m,n)=

∫∞

0

tm−^1

(1+t)m+n

dt (1.21)

Special funtions, properties and differential equations

Hermite functions:

Differential equation:

y′′− 2 xy′+ 2 ny= 0 (1.22)

whenn= 0 , 1 , 2 ,...then we get Hermite’s polynomialsHn(x)ofdegreen,given
by

Hn(x)=(−1)nex

2 dn

dxn

(

e−x

2 )

(Rodrigue’s formula)