In mathematics , the incomplete Fermi-Dirac integral , named after Enrico Fermi and Paul Dirac , for an index
j
{\displaystyle j}
b
{\displaystyle b}
F
j
(
x
,
b
)
=
d
e
f
1
Γ
(
j
+
1
)
∫
b
∞
t
j
e
t
−
x
+
1
d
t
{\displaystyle \operatorname {F} _{j}(x,b){\overset {\mathrm {def} }{=}}{\frac {1}{\Gamma (j+1)}}\int _{b}^{\infty }\!{\frac {t^{j}}{e^{t-x}+1}}\;\mathrm {d} t}
Its derivative is
d
d
x
F
j
(
x
,
b
)
=
F
j
−
1
(
x
,
b
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {F} _{j}(x,b)=\operatorname {F} _{j-1}(x,b)}
and this derivative relationship may be used to find the value of the incomplete Fermi-Dirac integral for non-positive indices
j
{\displaystyle j}
This is an alternate definition of the incomplete polylogarithm , since:
F
j
(
x
,
b
)
=
1
Γ
(
j
+
1
)
∫
b
∞
t
j
e
t
−
x
+
1
d
t
=
1
Γ
(
j
+
1
)
∫
b
∞
t
j
e
t
e
x
+
1
d
t
=
−
1
Γ
(
j
+
1
)
∫
b
∞
t
j
e
t
−
e
x
−
1
d
t
=
−
Li
j
+
1
(
b
,
−
e
x
)
{\displaystyle \operatorname {F} _{j}(x,b)={\frac {1}{\Gamma (j+1)}}\int _{b}^{\infty }\!{\frac {t^{j}}{e^{t-x}+1}}\;\mathrm {d} t={\frac {1}{\Gamma (j+1)}}\int _{b}^{\infty }\!{\frac {t^{j}}{\displaystyle {\frac {e^{t}}{e^{x}}}+1}}\;\mathrm {d} t=-{\frac {1}{\Gamma (j+1)}}\int _{b}^{\infty }\!{\frac {t^{j}}{\displaystyle {\frac {e^{t}}{-e^{x}}}-1}}\;\mathrm {d} t=-\operatorname {Li} _{j+1}(b,-e^{x})}
Which can be used to prove the identity:
F
j
(
x
,
b
)
=
−
∑
n
=
1
∞
(
−
1
)
n
n
j
+
1
Γ
(
j
+
1
,
n
b
)
Γ
(
j
+
1
)
e
n
x
{\displaystyle \operatorname {F} _{j}(x,b)=-\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{j+1}}}{\frac {\Gamma (j+1,nb)}{\Gamma (j+1)}}e^{nx}}
where
Γ
(
s
)
{\displaystyle \Gamma (s)}
gamma function and
Γ
(
s
,
y
)
{\displaystyle \Gamma (s,y)}
incomplete gamma function . Since
Γ
(
s
,
0
)
=
Γ
(
s
)
{\displaystyle \Gamma (s,0)=\Gamma (s)}
F
j
(
x
,
0
)
=
F
j
(
x
)
{\displaystyle \operatorname {F} _{j}(x,0)=\operatorname {F} _{j}(x)}
where
F
j
(
x
)
{\displaystyle \operatorname {F} _{j}(x)}
complete Fermi-Dirac integral .
Special values
The closed form of the function exists for
j
=
0
{\displaystyle j=0}
F
0
(
x
,
b
)
=
ln
(
1
+
e
x
−
b
)
−
(
b
−
x
)
{\displaystyle \operatorname {F} _{0}(x,b)=\ln \!{\big (}1+e^{x-b}{\big )}-(b-x)}
See also
References
^ Guano, Michele (1995). "Algorithm 745: computation of the complete and incomplete Fermi-Dirac integral" . ACM Transactions on Mathematical Software . 21 (3): 221–232. doi :10.1145/210089.210090 . Retrieved 26 June 2024.
External links
Categories :
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