Line 92:
Line 92:
:<math>\psi\left(\frac{m}{k}\right) = -\gamma -\ln(2k)
:<math>\psi\left(\frac{m}{k}\right) = -\gamma -\ln(2k)
-\frac{\pi}{2}\cot\left(\frac{m\pi}{k}\right)
-\frac{\pi}{2}\cot\left(\frac{m\pi}{k}\right)
+2\sum_{n=1}^{\lfloor (k-1)/2\rfloor}
+2\sum_{n=1}^{\lceil (k-1)/2\rceil }
\cos\left(\frac{2\pi nm}{k} \right)
\cos\left(\frac{2\pi nm}{k} \right)
\ln\left(\sin\left(\frac{n\pi}{k}\right)\right)
\ln\left(\sin\left(\frac{n\pi}{k}\right)\right)
Digamma function
ψ
(
s
)
{\displaystyle \psi (s)}
complex plane . The color of a point
s
{\displaystyle s}
ψ
(
s
)
{\displaystyle \psi (s)}
argument . In mathematics , the digamma function is defined as the logarithmic derivative of the gamma function :
ψ
(
x
)
=
d
d
x
ln
Γ
(
x
)
=
Γ
′
(
x
)
Γ
(
x
)
.
{\displaystyle \psi (x)={\frac {d}{dx}}\ln {\Gamma (x)}={\frac {\Gamma '(x)}{\Gamma (x)}}.}
It is the first of the polygamma functions .
Relation to harmonic numbers
The digamma function , often denoted also as ψ0 (x ), ψ0 (x ) or
ϝ
{\displaystyle \digamma }
digamma ), is related to the harmonic numbers in that
ψ
(
n
)
=
H
n
−
1
−
γ
{\displaystyle \psi (n)=H_{n-1}-\gamma \!}
where H n n 'th harmonic number, and γ is the Euler-Mascheroni constant . For half-integer values, it may be expressed as
ψ
(
n
+
1
2
)
=
−
γ
−
2
ln
2
+
∑
k
=
1
n
2
2
k
−
1
{\displaystyle \psi \left(n+{\frac {1}{2}}\right)=-\gamma -2\ln 2+\sum _{k=1}^{n}{\frac {2}{2k-1}}}
Integral representations
It has the integral representation
ψ
(
x
)
=
∫
0
∞
(
e
−
t
t
−
e
−
x
t
1
−
e
−
t
)
d
t
{\displaystyle \psi (x)=\int _{0}^{\infty }\left({\frac {e^{-t}}{t}}-{\frac {e^{-xt}}{1-e^{-t}}}\right)\,dt}
This may be written as
ψ
(
s
+
1
)
=
−
γ
+
∫
0
1
1
−
x
s
1
−
x
d
x
{\displaystyle \psi (s+1)=-\gamma +\int _{0}^{1}{\frac {1-x^{s}}{1-x}}dx}
which follows from Euler's integral formula for the harmonic numbers.
Taylor series
The digamma has a rational zeta series , given by the Taylor series at z =1. This is
ψ
(
z
+
1
)
=
−
γ
−
∑
k
=
1
∞
ζ
(
k
+
3
)
(
−
z
)
k
{\displaystyle \psi (z+1)=-\gamma -\sum _{k=1}^{\infty }\zeta (k+3)\;(-z)^{k}}
which converges for |z |<1. Here,
ζ
(
n
)
{\displaystyle \zeta (n)}
Riemann zeta function . This series is easily derived from the corresponding Taylor's series for the Hurwitz zeta function .
Newton series
The Newton series for the digamma follows from Euler's integral formula:
ψ
(
s
+
1
)
=
−
γ
−
∑
k
=
1
∞
(
−
1
)
k
k
(
s
k
)
{\displaystyle \psi (s+1)=-\gamma -\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}{s \choose k}}
where
(
s
k
)
{\displaystyle \textstyle {s \choose k}}
binomial coefficient .
The digamma function satisfies a reflection formula similar to that of the Gamma function ,
ψ
(
1
−
x
)
−
ψ
(
x
)
=
π
cot
(
π
x
)
{\displaystyle \psi (1-x)-\psi (x)=\pi \,\!\cot {\left(\pi x\right)}}
The digamma function satisfies the recurrence relation
ψ
(
x
+
1
)
=
ψ
(
x
)
+
1
x
{\displaystyle \psi (x+1)=\psi (x)+{\frac {1}{x}}}
Thus, it can be said to "telescope" 1/x, for one has
Δ
[
ψ
]
(
x
)
=
1
x
{\displaystyle \Delta [\psi ](x)={\frac {1}{x}}}
where Δ is the forward difference operator . This satisfies the recurrence relation of a partial sum of the harmonic series , thus implying the formula
ψ
(
n
)
=
H
n
−
1
−
γ
{\displaystyle \psi (n)\ =\ H_{n-1}-\gamma }
where
γ
{\displaystyle \gamma }
Euler-Mascheroni constant .
More generally, one has
ψ
(
x
+
1
)
=
−
γ
+
∑
k
=
1
∞
(
1
k
−
1
x
+
k
)
{\displaystyle \psi (x+1)=-\gamma +\sum _{k=1}^{\infty }\left({\frac {1}{k}}-{\frac {1}{x+k}}\right)}
Gaussian sum
The digamma has a Gaussian sum of the form
−
1
π
k
∑
n
=
1
k
sin
(
2
π
n
m
k
)
ψ
(
n
k
)
=
ζ
(
0
,
m
k
)
=
−
B
1
(
m
k
)
=
1
2
−
m
k
{\displaystyle {\frac {-1}{\pi k}}\sum _{n=1}^{k}\sin \left({\frac {2\pi nm}{k}}\right)\psi \left({\frac {n}{k}}\right)=\zeta \left(0,{\frac {m}{k}}\right)=-B_{1}\left({\frac {m}{k}}\right)={\frac {1}{2}}-{\frac {m}{k}}}
for integers
0
<
m
<
k
{\displaystyle 0<m<k}
s ,q ) is the Hurwitz zeta function and
B
n
(
x
)
{\displaystyle B_{n}(x)}
Bernoulli polynomial . A special case of the multiplication theorem is
∑
n
=
1
k
ψ
(
n
k
)
=
−
k
(
γ
+
log
k
)
,
{\displaystyle \sum _{n=1}^{k}\psi \left({\frac {n}{k}}\right)=-k(\gamma +\log k),}
and a neat generalization of this is
∑
p
=
0
q
−
1
ψ
(
a
+
p
/
q
)
=
q
(
ψ
(
q
a
)
−
ln
(
q
)
)
,
{\displaystyle \sum _{p=0}^{q-1}\psi (a+p/q)=q(\psi (qa)-\ln(q)),}
in which it is assumed that q is a natural number, and that 1-qa is not.
For positive integers m and k (with m < k ), the digamma function may be expressed in terms of elementary functions as
ψ
(
m
k
)
=
−
γ
−
ln
(
2
k
)
−
π
2
cot
(
m
π
k
)
+
2
∑
n
=
1
⌈
(
k
−
1
)
/
2
⌉
cos
(
2
π
n
m
k
)
ln
(
sin
(
n
π
k
)
)
{\displaystyle \psi \left({\frac {m}{k}}\right)=-\gamma -\ln(2k)-{\frac {\pi }{2}}\cot \left({\frac {m\pi }{k}}\right)+2\sum _{n=1}^{\lceil (k-1)/2\rceil }\cos \left({\frac {2\pi nm}{k}}\right)\ln \left(\sin \left({\frac {n\pi }{k}}\right)\right)}
Special values
The digamma function has the following special values:
ψ
(
1
)
=
−
γ
{\displaystyle \psi (1)=-\gamma \,\!}
ψ
(
1
2
)
=
−
2
ln
2
−
γ
{\displaystyle \psi \left({\frac {1}{2}}\right)=-2\ln {2}-\gamma }
ψ
(
1
3
)
=
−
π
2
3
−
3
2
ln
3
−
γ
{\displaystyle \psi \left({\frac {1}{3}}\right)=-{\frac {\pi }{2{\sqrt {3}}}}-{\frac {3}{2}}\ln {3}-\gamma }
ψ
(
1
4
)
=
−
π
2
−
3
ln
2
−
γ
{\displaystyle \psi \left({\frac {1}{4}}\right)=-{\frac {\pi }{2}}-3\ln {2}-\gamma }
ψ
(
1
6
)
=
−
π
2
3
−
2
ln
2
−
3
2
ln
(
3
)
−
γ
{\displaystyle \psi \left({\frac {1}{6}}\right)=-{\frac {\pi }{2}}{\sqrt {3}}-2\ln {2}-{\frac {3}{2}}\ln(3)-\gamma }
ψ
(
1
8
)
=
−
π
2
−
4
ln
2
−
1
2
{
π
+
ln
(
2
+
2
)
−
ln
(
2
−
2
)
}
−
γ
{\displaystyle \psi \left({\frac {1}{8}}\right)=-{\frac {\pi }{2}}-4\ln {2}-{\frac {1}{\sqrt {2}}}\left\{\pi +\ln(2+{\sqrt {2}})-\ln(2-{\sqrt {2}})\right\}-\gamma }
See also
References
External links
Cephes - C and C++ language special functions math library
={\frac {1}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f937d04ca5581f9bf986c18bf170bdc9b376cc8)
