Γ
1
2
þ m
¼ m À
1
2
Γ m À
1
2
¼ m À
1
2
m À
3
2
Γ m À
3
2
Á Á Á ¼
¼ m À
1
2
m À
3
2
Á Á Á
1
2
Γ
1
2
¼ 2
Àm 2m À 1
ð
Þ2m À 3
ð
ÞÁÁÁ3 Á 1 Á Γ
1
2
¼ 2
Àm
2m À 1
ð
Þ!
2
mÀ1 m À 1
ð
Þ!
Γ
1
2
¼ 2
À2m 2m
ð Þ!
m!
Γ
1
2
:
ð3:195Þ
Notice that (3.195) still holds even if m ¼ 0. Inserting n À k into m of (3.195), we
get
Γ
1
2
þ n À k
¼ 2
À2 nÀk
ð
Þ 2n À 2k
ð
Þ !
n À k
ð
Þ!
Γ
1
2
:
ð3:196Þ
Replacing Γ
1
2 þ n À k
À
Á
of (3.193) with RHS of the above equation, (3.194) will
follow. A gamma function Γ
1
2
À Á
often appears in mathematical physics. According to
(3.185), we have
Γ
1
2
¼ 2
Z 1
0
e
Àu
2 du ¼
ffiffiffi
π
p :
For the derivation of the above definite integral, see (2.86) of Sect. 2.4. From
(3.184), we also have
Γ 1
ð Þ ¼ 1:
In relation of the discussion of Sect. 3.5, let us derive an important formula about
Legendre polynomials. From (3.180) and (3.192), we get
1 À 2tx þ t
2
À
Á À1=2
X 1
n¼0
P n x
ð Þt
n
:
ð3:197Þ
Assuming jt j < 1, when we put x ¼ 1 in (3.197), we have
1 À 2tx þ t
2
À
Á À
1
2
¼
1
1 À t
¼
X 1
n¼0
t
n
¼
X 1
n¼0
P n 1
ð Þt
n
:
ð3:198Þ
Comparing individual coefficients of t
n in (3.198), we get
P n 1
ð Þ ¼ 1:
See the related parts of Sect. 3.5.
Now, we are in the position to determine the constant in (3.179). Differentiating
(3.194) m times, we have
98
3 Hydrogen-Like Atoms
1
2
þ m
¼ m À
1
2
Γ m À
1
2
¼ m À
1
2
m À
3
2
Γ m À
3
2
Á Á Á ¼
¼ m À
1
2
m À
3
2
Á Á Á
1
2
Γ
1
2
¼ 2
Àm 2m À 1
ð
Þ2m À 3
ð
ÞÁÁÁ3 Á 1 Á Γ
1
2
¼ 2
Àm
2m À 1
ð
Þ!
2
mÀ1 m À 1
ð
Þ!
Γ
1
2
¼ 2
À2m 2m
ð Þ!
m!
Γ
1
2
:
ð3:195Þ
Notice that (3.195) still holds even if m ¼ 0. Inserting n À k into m of (3.195), we
get
Γ
1
2
þ n À k
¼ 2
À2 nÀk
ð
Þ 2n À 2k
ð
Þ !
n À k
ð
Þ!
Γ
1
2
:
ð3:196Þ
Replacing Γ
1
2 þ n À k
À
Á
of (3.193) with RHS of the above equation, (3.194) will
follow. A gamma function Γ
1
2
À Á
often appears in mathematical physics. According to
(3.185), we have
Γ
1
2
¼ 2
Z 1
0
e
Àu
2 du ¼
ffiffiffi
π
p :
For the derivation of the above definite integral, see (2.86) of Sect. 2.4. From
(3.184), we also have
Γ 1
ð Þ ¼ 1:
In relation of the discussion of Sect. 3.5, let us derive an important formula about
Legendre polynomials. From (3.180) and (3.192), we get
1 À 2tx þ t
2
À
Á À1=2
X 1
n¼0
P n x
ð Þt
n
:
ð3:197Þ
Assuming jt j < 1, when we put x ¼ 1 in (3.197), we have
1 À 2tx þ t
2
À
Á À
1
2
¼
1
1 À t
¼
X 1
n¼0
t
n
¼
X 1
n¼0
P n 1
ð Þt
n
:
ð3:198Þ
Comparing individual coefficients of t
n in (3.198), we get
P n 1
ð Þ ¼ 1:
See the related parts of Sect. 3.5.
Now, we are in the position to determine the constant in (3.179). Differentiating
(3.194) m times, we have
98
3 Hydrogen-Like Atoms