Complete Fermi–Dirac integral

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Digamma function $\psi (s)$ in the complex plane. The color of a point $s$ encodes the value of $\psi (s)$. Strong colors denote values close to zero and hue encodes the value's argument.

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:

$\psi (x)={\frac {d}{dx}}\ln {\Gamma (x)}={\frac {\Gamma '(x)}{\Gamma (x)}}.$

It is the first of the polygamma functions.

Relation to harmonic numbers

The digamma function, often denoted also as ψ₀(x), ψ⁰(x) or $\digamma$ (after the shape of the archaic Greek letter Ϝ digamma), is related to the harmonic numbers in that

$\psi (n)=H_{n-1}-\gamma \!$

where H_n is the n^th harmonic number, and γ is the Euler-Mascheroni constant. For half-integer values, it may be expressed as

$\psi \left(n+{\frac {1}{2}}\right)=-\gamma -2\ln 2+\sum _{k=1}^{n}{\frac {2}{2k-1}}$

Integral representations

If the real part of $x$ is positive then the digamma function has the following integral representation

$\psi (x)=\int _{0}^{\infty }\left({\frac {e^{-t}}{t}}-{\frac {e^{-xt}}{1-e^{-t}}}\right)\,dt$.

This may be written as

$\psi (s+1)=-\gamma +\int _{0}^{1}{\frac {1-x^{s}}{1-x}}dx$

which follows from Euler's integral formula for the harmonic numbers.

Series formula

Digamma can be computed in the complex plane outside negative integers (Abramowitz and Stegun 6.3.16), using

$\psi (z+1)=-\gamma +\sum _{n=1}^{\infty }{\frac {z}{n(n+z)}}\qquad z\neq -1,-2,-3,\ldots$

or

$\psi (z)=-\gamma +\sum _{n=0}^{\infty }{\frac {z-1}{(n+1)(n+z)}}=-\gamma +\sum _{n=0}^{\infty }\left({\frac {1}{n+1}}-{\frac {1}{n+z}}\right)\qquad z\neq 0,-1,-2,-3,\ldots$

This can be utilized to evaluate infinite sums of rational functions, i.e., $\sum _{n=0}^{\infty }u_{n}=\sum _{n=0}^{\infty }{\frac {p(n)}{q(n)}}$, where p(n) and q(n) are polynomials of n.

Performing partial fraction on u_n in the complex field, in the case when all roots of q(n) are simple roots,

$u_{n}={\frac {p(n)}{q(n)}}=\sum _{k=1}^{m}{\frac {a_{k}}{n+b_{k}}}.$

For the series to converge,

$\lim _{n\to \infty }nu_{n}=0,$

or otherwise the series will be greater than harmonic series and thus diverges.

Hence

$\sum _{k=1}^{m}a_{k}=0,$

and

$\sum _{n=0}^{\infty }u_{n}=\sum _{n=0}^{\infty }\sum _{k=1}^{m}{\frac {a_{k}}{n+b_{k}}}=\sum _{n=0}^{\infty }\sum _{k=1}^{m}a_{k}\left({\frac {1}{n+b_{k}}}-{\frac {1}{n+1}}\right)=$

$=\sum _{k=1}^{m}\left(a_{k}\sum _{n=0}^{\infty }\left({\frac {1}{n+b_{k}}}-{\frac {1}{n+1}}\right)\right)=-\sum _{k=1}^{m}a_{k}\left(\psi (b_{k})+\gamma \right)=-\sum _{k=1}^{m}a_{k}\psi (b_{k}).$

With the series expansion of higher rank polygamma function a generalized formula can be given as

$\sum _{n=0}^{\infty }u_{n}=\sum _{n=0}^{\infty }\sum _{k=1}^{m}{\frac {a_{k}}{(n+b_{k})^{r_{k}}}}=\sum _{k=1}^{m}{\frac {(-1)^{r_{k}}}{(r_{k}-1)!}}a_{k}\psi ^{(r_{k}-1)}(b_{k}),$

provided the series on the left converges.

Taylor series

The digamma has a rational zeta series, given by the Taylor series at z=1. This is

$\psi (z+1)=-\gamma -\sum _{k=1}^{\infty }\zeta (k+1)\;(-z)^{k}$,

which converges for |z|<1. Here, $\zeta (n)$ is the 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:

$\psi (s+1)=-\gamma -\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}{s \choose k}$

where $\textstyle {s \choose k}$ is the binomial coefficient.

Reflection formula

The digamma function satisfies a reflection formula similar to that of the Gamma function,

$\psi (1-x)-\psi (x)=\pi \,\!\cot {\left(\pi x\right)}$

Recurrence formula and characterization

The digamma function satisfies the recurrence relation

$\psi (x+1)=\psi (x)+{\frac {1}{x}}.$

Thus, it can be said to "telescope" 1/x, for one has

$\Delta [\psi ](x)={\frac {1}{x}}$

where Δ is the forward difference operator. This satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula

$\psi (n)\ =\ H_{n-1}-\gamma$

where $\gamma \,$ is the Euler-Mascheroni constant.

More generally, one has

$\psi (x+1)=-\gamma +\sum _{k=1}^{\infty }\left({\frac {1}{k}}-{\frac {1}{x+k}}\right).$

Actually, $\psi$ is the only solution of the functional equation $F(x+1)=F(x)+{\frac {1}{x}}$ that is monotone on $\mathbb {R} ^{+}$ and satisfies $F(1)=-\gamma$. This fact follows immediately from the uniqueness of the $\Gamma$ function given its recurrence equation and convexity-restriction. This implies the useful difference equation :

$\psi (x+N)-\psi (x)=\sum _{k=0}^{N-1}{\frac {1}{x+k}}$

Gaussian sum

The digamma has a Gaussian sum of the form

${\frac {-1}{\pi k}}\sum _{n=1}^{k}\sin \left({\frac {2\pi nm}{k}}\right)\psi \left({\frac {n}{k}}\right)=\zeta \left(0,{\frac {m}{k}}\right)=-B_{1}\left({\frac {m}{k}}\right)={\frac {1}{2}}-{\frac {m}{k}}$

for integers $0<m<k$. Here, ζ(s,q) is the Hurwitz zeta function and $B_{n}(x)$ is a Bernoulli polynomial. A special case of the multiplication theorem is

$\sum _{n=1}^{k}\psi \left({\frac {n}{k}}\right)=-k(\gamma +\log k),$

and a neat generalization of this is

$\sum _{p=0}^{q-1}\psi (a+p/q)=q(\psi (qa)-\log(q)),$

where q must be a natural number, but 1-qa not.

Gauss's digamma theorem

For positive integers m and k (with m < k), the digamma function may be expressed in finite many terms of elementary functions as

$\psi \left({\frac {m}{k}}\right)=-\gamma -\ln(2k)-{\frac {\pi }{2}}\cot \left({\frac {m\pi }{k}}\right)+2\sum _{n=1}^{\lfloor (k-1)/2\rfloor }\cos \left({\frac {2\pi nm}{k}}\right)\ln \left(\sin \left({\frac {n\pi }{k}}\right)\right)$

and because of its recurrence equation for all rational arguments.

Computation and approximation

According to the Euler Maclaurin formula applied for $\sum _{n=1}^{x}{\tfrac {1}{n}}$ the digamma function for x, also a real number, can be approximated by

$\psi (x)=\ln(x)-{\frac {1}{2x}}-{\frac {1}{12x^{2}}}+{\frac {1}{120x^{4}}}-{\frac {1}{252x^{6}}}+{\frac {1}{240x^{8}}}-{\frac {5}{660x^{10}}}+{\frac {691}{32760x^{12}}}-{\frac {1}{12x^{14}}}+O\left({\frac {1}{x^{16}}}\right)$

which is the beginning of the asymptotical expansion of $\psi (x)$. The full asymptotic series of this expansions is

$\psi (x)=\ln(x)-{\frac {1}{2x}}+\sum _{n=1}^{\infty }{\frac {\zeta (1-2n)}{x^{2n}}}=\ln(x)-{\frac {1}{2x}}-\sum _{n=1}^{\infty }{\frac {B_{2n}}{2n\,x^{2n}}}$

where $B_{k}$ is the kth Bernoulli number and $\zeta$ is the Riemann zeta function. Although the infinite sum converges for no x, this expansion becomes more accurate for larger values of x and any finite partial sum cut off from the full series. To compute $\psi (x)$ for small x, the recurrence relation

$\psi (x+1)={\frac {1}{x}}+\psi (x)$

can be used to shift the value of x to a higher value. Beal suggests using the above recurrence to shift x to a value greater than 6 and then applying the above expansion with terms above $x^{14}$ cut off, which yields "more than enough precision" (at least 12 digits except near the zeroes).

$\psi (x)\in [\ln(x-1),\ln x]$

$\exp(\psi (x))\approx {\begin{cases}{\frac {x^{2}}{2}}&:x\in [0,1]\\x-{\frac {1}{2}}&:x>1\end{cases}}$

From the above asymptotic series for $\psi$ you can derive asymptotic series for $\exp \circ \,\psi$ that contain only rational functions and constants. The first series matches the overall behaviour of $\exp \circ \,\psi$ well, that is, it behaves asymptotically identically for large arguments and has a zero of unbounded multiplicity at the origin, too. It can be considered a Taylor expansion of $\exp(-\psi (1/y))$ at $y=0$.

${\frac {1}{\exp(\psi (x))}}={\frac {1}{x}}+{\frac {1}{2\cdot x^{2}}}+{\frac {5}{4\cdot 3!\cdot x^{3}}}+{\frac {3}{2\cdot 4!\cdot x^{4}}}+{\frac {47}{48\cdot 5!\cdot x^{5}}}-{\frac {5}{16\cdot 6!\cdot x^{6}}}+\dots$

The other expansion is more precise for large arguments and saves computing terms of even order.

$\exp(\psi (x+{\tfrac {1}{2}}))=x+{\frac {1}{4!\cdot x}}-{\frac {37}{8\cdot 6!\cdot x^{3}}}+{\frac {10313}{72\cdot 8!\cdot x^{5}}}-{\frac {5509121}{384\cdot 10!\cdot x^{7}}}+O\left({\frac {1}{x^{9}}}\right)\quad {\mbox{for }}x>1$

Special values

The digamma function has values in closed form for rational numbers, as a result of Gauss's digamma theorem. Some are listed below:

$\psi (1)=-\gamma \,\!$

$\psi \left({\frac {1}{2}}\right)=-2\ln {2}-\gamma$

$\psi \left({\frac {1}{3}}\right)=-{\frac {\pi }{2{\sqrt {3}}}}-{\frac {3}{2}}\ln {3}-\gamma$

$\psi \left({\frac {1}{4}}\right)=-{\frac {\pi }{2}}-3\ln {2}-\gamma$

$\psi \left({\frac {1}{6}}\right)=-{\frac {\pi }{2}}{\sqrt {3}}-2\ln {2}-{\frac {3}{2}}\ln(3)-\gamma$

$\psi \left({\frac {1}{8}}\right)=-{\frac {\pi }{2}}-4\ln {2}-{\frac {1}{\sqrt {2}}}\left\{\pi +\ln(2+{\sqrt {2}})-\ln(2-{\sqrt {2}})\right\}-\gamma$

The roots of the digamma function are the saddle points of the complex-valued gamma function. Thus they lie all on the real axis. The only one on the positive real axis is the unique minimum of the real-valued gamma function on $\mathbb {R} ^{+}$ at $x_{0}=1.461632144968\ldots$. All others occur single between the pols on the negative axis: $x_{1}=-0.504083008\ldots ,x_{2}=-1.573498473\ldots ,x_{3}=-2.610720868\ldots ,x_{4}=-3.635293366\ldots ,\ldots$. Already 1881 Hermite observed that $x_{n}=-n+{\frac {1}{\ln n}}+o\left({\frac {1}{\ln ^{2}n}}\right)$ holds asymptotically . A better approximation of the location of the roots is given by

$x_{n}\approx -n+{\frac {1}{\pi }}\arctan \left({\frac {\pi }{\ln n}}\right)\qquad n\geq 2$

and using a further term it becomes still better

$x_{n}\approx -n+{\frac {1}{\pi }}\arctan \left({\frac {\pi }{\ln n+{\frac {1}{8n}}}}\right)\qquad n\geq 1$

which both spring off the reflection formula via $0=\psi (1-x_{n})=\psi (x_{n})+{\frac {\pi }{\tan(\pi x_{n})}}$ and substituting $\psi (x_{n})$ by its not convergent asymptotic expansion. The correct 2nd term of this expansion is of course ${\tfrac {1}{2n}}$, where the given one works good to approximate roots with small index n.

Regularization

the Digamma function appears in the regularization of divergent integrals $\int _{0}^{\infty }{\frac {dx}{x+a}}$, this integral can be approximated by a divergent general Harmonic series, but the following value can be attached to the series $\sum _{n=0}^{\infty }{\frac {1}{n+a}}=-\psi (a)$

See also

• Polygamma function
• Trigamma function
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External links

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