Γ z
ð Þ ¼ 2
Z 1
0
e
Àu
2 u
2zÀ1 du Re z > 0
ð
Þ :
ð3:185Þ
Note that the above expression is associated with the following fundamental
feature of the gamma functions:
Γ z þ 1
ð
Þ¼zΓ z
ð Þ,
ð3:186Þ
where z is any complex number.
Replacing x with Àt(2x À t) and rewriting (3.183), we have
1 À 2tx þ t
2
À
Á Àλ ¼
X 1
m¼0
Γ Àλ þ 1
ð
Þ
m!Γ Àλ À m þ 1
ð
Þ
Àt
ð Þ
m 2x À t
ð
Þ
m :
ð3:187Þ
Assuming that x is a real number belonging to an interval [À1, 1], (3.187) holds
with t satisfying jt j < 1 [8]. The discussion is as follows: When x satisfies the above
condition, solving 1 À 2tx + t
2
¼ 0 we get solution t Æ such that
t Æ ¼ x Æ i
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À x 2
p
:
Defining r as
r min jt þ j, jt À j
f
g ,
(1 À 2tx + t
2 )
Àλ , regarded as a function of t, is analytic in the disk |t| < r. But, we
have
t Æ
j j ¼ 1:
Thus, (1 À 2tx + t
2 )
Àλ is analytic within the disk jt j < 1 and, hence, it can be
expanded in a Taylor’s series (see Chap. 6).
Continuing the calculation of (3.187), we have
1 À 2tx þ t
2
À
Á Àλ
¼
X 1
m¼0
Γ Àλ þ 1
ð
Þ
m!Γ Àλ À m þ 1
ð
Þ
À1
ð Þ
m t
m
X m
k¼0
m!
k! m À k
ð
Þ!
2
mÀk x
mÀk
À1
ð Þ
k t
k
!
¼
X 1
m¼0
X m
k¼0
À1
ð Þ
mþk
k! m À k
ð
Þ!
Γ Àλ þ 1
ð
Þ
Γ Àλ À m þ 1
ð
Þ
2
mÀk x
mÀk t
mþk
¼
X 1
m¼0
X m
k¼0
À1
ð Þ
mþk
k! m À k
ð
Þ!
À1
ð Þ
m Γ λ þ m
ð
Þ
Γ λ
ð Þ
2
mÀk x
mÀk t
mþk ,
ð3:188Þ
96
3 Hydrogen-Like Atoms
ð Þ ¼ 2
Z 1
0
e
Àu
2 u
2zÀ1 du Re z > 0
ð
Þ :
ð3:185Þ
Note that the above expression is associated with the following fundamental
feature of the gamma functions:
Γ z þ 1
ð
Þ¼zΓ z
ð Þ,
ð3:186Þ
where z is any complex number.
Replacing x with Àt(2x À t) and rewriting (3.183), we have
1 À 2tx þ t
2
À
Á Àλ ¼
X 1
m¼0
Γ Àλ þ 1
ð
Þ
m!Γ Àλ À m þ 1
ð
Þ
Àt
ð Þ
m 2x À t
ð
Þ
m :
ð3:187Þ
Assuming that x is a real number belonging to an interval [À1, 1], (3.187) holds
with t satisfying jt j < 1 [8]. The discussion is as follows: When x satisfies the above
condition, solving 1 À 2tx + t
2
¼ 0 we get solution t Æ such that
t Æ ¼ x Æ i
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À x 2
p
:
Defining r as
r min jt þ j, jt À j
f
g ,
(1 À 2tx + t
2 )
Àλ , regarded as a function of t, is analytic in the disk |t| < r. But, we
have
t Æ
j j ¼ 1:
Thus, (1 À 2tx + t
2 )
Àλ is analytic within the disk jt j < 1 and, hence, it can be
expanded in a Taylor’s series (see Chap. 6).
Continuing the calculation of (3.187), we have
1 À 2tx þ t
2
À
Á Àλ
¼
X 1
m¼0
Γ Àλ þ 1
ð
Þ
m!Γ Àλ À m þ 1
ð
Þ
À1
ð Þ
m t
m
X m
k¼0
m!
k! m À k
ð
Þ!
2
mÀk x
mÀk
À1
ð Þ
k t
k
!
¼
X 1
m¼0
X m
k¼0
À1
ð Þ
mþk
k! m À k
ð
Þ!
Γ Àλ þ 1
ð
Þ
Γ Àλ À m þ 1
ð
Þ
2
mÀk x
mÀk t
mþk
¼
X 1
m¼0
X m
k¼0
À1
ð Þ
mþk
k! m À k
ð
Þ!
À1
ð Þ
m Γ λ þ m
ð
Þ
Γ λ
ð Þ
2
mÀk x
mÀk t
mþk ,
ð3:188Þ
96
3 Hydrogen-Like Atoms