In mathematics, the Lerch transcendent, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about a similar function in 1887. The Lerch transcendent, is given by:
:<math>\Phi(z, s, \alpha) = \sum_{n=0}^\infty
\frac { z^n} {(n+\alpha)^s}</math>.
It only converges for any real number <math>\alpha > 0</math>, where <math>|z| < 1</math>, or <math>\mathfrak{R}(s) > 1</math>, and <math>|z| = 1</math>.
Special cases
The Lerch transcendent is related to and generalizes various special functions.
The Lerch zeta function is given by:
:<math>L(\lambda, s, \alpha) = \sum_{n=0}^\infty
\frac { e^{2\pi i\lambda n {(n+\alpha)^s}=\Phi(e^{2\pi i\lambda}, s,\alpha)</math>
The Hurwitz zeta function is the special case
:<math>\zeta(s,\alpha) = \sum_{n=0}^\infty \frac{1}{(n+\alpha)^s} = \Phi(1,s,\alpha)</math>
The polylogarithm is another special case:
:<math>\lambda(s) = \sum_{k=0}^\infty \frac{1}{(2k+1)^s} = 2^{-s}\Phi(1,s,\tfrac12)</math>
The Legendre chi function:
:<math>\textrm{Ti}_s(z)= \sum_{k=0}^\infty \frac{(-1)^k z^{2k+1{(2k+1)^s}=\frac{z}{2^s}\Phi(-z^2,s,\tfrac12) </math>
The polygamma functions for positive integers n:
:<math>\psi^{(n)}(\alpha)= (-1)^{n+1} n!\Phi (1,n+1,\alpha)</math>
The Clausen function:
:<math>\text{Cl}_2(z)= \frac{ie^{-iz2 \Phi(e^{-iz},2,1)-\frac{ie^{iz2 \Phi(e^{iz},2,1)</math>
Integral representations
The Lerch transcendent has an integral representation:
:<math>
\Phi(z,s,a)=\frac{1}{\Gamma(s)}\int_0^\infty
\frac{t^{s-1}e^{-at{1-ze^{-t\,dt</math>
The proof is based on using the integral definition of the gamma function to write
:<math>\Phi(z,s,a)\Gamma(s)
= \sum_{n=0}^\infty \frac{z^n}{(n+a)^s} \int_0^\infty x^s e^{-x} \frac{dx}{x}
= \sum_{n=0}^\infty \int_0^\infty t^s z^n e^{-(n+a)t} \frac{dt}{t}</math>
and then interchanging the sum and integral. The resulting integral representation converges for <math>z \in \Complex \setminus [1,\infty),</math> Re(s) > 0, and Re(a) > 0. This analytically continues <math>\Phi(z,s,a)</math> to z outside the unit disk. The integral formula also holds if z = 1, Re(s) > 1, and Re(a) > 0; see Hurwitz zeta function.
A contour integral representation is given by
:<math>
\Phi(z,s,a)=-\frac{\Gamma(1-s)}{2\pi i} \int_C \frac{(-t)^{s-1}e^{-at{1-ze^{-t\,dt</math>
where C is a Hankel contour counterclockwise around the positive real axis, not enclosing any of the points <math>t = \log(z) + 2k\pi i</math> (for integer k) which are poles of the integrand. The integral assumes Re(a) > 0.
Other integral representations
A Hermite-like integral representation is given by
:<math>
\Phi(z,s,a)=
\frac{1}{2a^s}+
\int_0^\infty \frac{z^t}{(a+t)^s}\,dt+
\frac{2}{a^{s-1
\int_0^\infty
\frac{\sin(s\arctan(t)-ta\log(z))}{(1+t^2)^{s/2}(e^{2\pi at}-1)}\,dt
</math>
for
:<math>\Re(a)>0\wedge |z|<1 </math>
and
:<math>
\Phi(z,s,a)=\frac{1}{2a^s}+
\frac{\log^{s-1}(1/z)}{z^a}\Gamma(1-s,a\log(1/z))+
\frac{2}{a^{s-1
\int_0^\infty
\frac{\sin(s\arctan(t)-ta\log(z))}{(1+t^2)^{s/2}(e^{2\pi at}-1)}\,dt
</math>
for
:<math>\Re(a)>0. </math>
Similar representations include
:<math>
\Phi(z,s,a)= \frac{1}{2a^s} + \int_{0}^{\infty}\frac{\cos(t\log z)\sin\Big(s\arctan\tfrac{t}{a}\Big) - \sin(t\log z)\cos\Big(s\arctan\tfrac{t}{a}\Big)}{\big(a^2 + t^2\big)^{\frac{s}{2 \tanh\pi t }\,dt,
</math>
and
:<math>\Phi(-z,s,a)= \frac{1}{2a^s} + \int_{0}^{\infty}\frac{\cos(t\log z)\sin\Big(s\arctan\tfrac{t}{a}\Big) - \sin(t\log z)\cos\Big(s\arctan\tfrac{t}{a}\Big)}{\big(a^2 + t^2\big)^{\frac{s}{2 \sinh\pi t }\,dt,</math>
holding for positive z (and more generally wherever the integrals converge). Furthermore,
:<math>\Phi(e^{i\varphi},s,a)=L\big(\tfrac{\varphi}{2\pi}, s, a\big)= \frac{1}{a^s} + \frac{1}{2\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-at}\big(e^{i\varphi}-e^{-t}\big)}{\cosh{t}-\cos{\varphi\,dt,</math>
The last formula is also known as Lipschitz formula.
Identities
For λ rational, the summand is a root of unity, and thus <math>L(\lambda, s, \alpha)</math> may be expressed as a finite sum over the Hurwitz zeta function. Suppose <math display="inline">\lambda = \frac{p}{q}</math> with <math>p, q \in \Z</math> and <math>q > 0</math>. Then <math>z = \omega = e^{2 \pi i \frac{p}{q</math> and <math>\omega^q = 1</math>.
:<math>\Phi(\omega, s, \alpha) = \sum_{n=0}^\infty
\frac {\omega^n} {(n+\alpha)^s} = \sum_{m=0}^{q-1} \sum_{n=0}^\infty \frac {\omega^{qn + m{(qn + m + \alpha)^s} = \sum_{m=0}^{q-1} \omega^m q^{-s} \zeta \left( s,\frac{m + \alpha}{q} \right) </math>
Various identities include:
:<math>\Phi(z,s,a)=z^n \Phi(z,s,a+n) + \sum_{k=0}^{n-1} \frac {z^k}{(k+a)^s}</math>
and
:<math>\Phi(z,s-1,a)=\left(a+z\frac{\partial}{\partial z}\right) \Phi(z,s,a)</math>
and
:<math>\Phi(z,s+1,a)=-\frac{1}{s}\frac{\partial}{\partial a} \Phi(z,s,a).</math>
Series representations
A series representation for the Lerch transcendent is given by
:<math>\Phi(z,s,q)=\frac{1}{1-z}
\sum_{n=0}^\infty \left(\frac{-z}{1-z} \right)^n
\sum_{k=0}^n (-1)^k \binom{n}{k} (q+k)^{-s}.</math>
(Note that <math>\tbinom{n}{k}</math> is a binomial coefficient.)
The series is valid for all s, and for complex z with Re(z)<1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function.
A Taylor series in the first parameter was given by Arthur Erdélyi. It may be written as the following series, which is valid for
:<math>\left|\log(z)\right| < 2 \pi;s\neq 1,2,3,\dots; a\neq 0,-1,-2,\dots</math>
:<math>
\Phi(z,s,a)=z^{-a}\left[\Gamma(1-s)\left(-\log (z)\right)^{s-1}
+\sum_{k=0}^\infty \zeta(s-k,a)\frac{\log^k (z)}{k!}\right]
</math>
If n is a positive integer, then
:<math>
\Phi(z,n,a)=z^{-a}\left\{
\sum_+
\frac{e^{2\pi ika}\Gamma(1-s,a(2\pi ik-\log(z)))}{(2\pi ik-\log(z))^{1-s
</math>
for <math>|a|<1;\Re(s)<0.</math>
The representation as a generalized hypergeometric function is
:<math>
\Phi(z,s,\alpha)=\frac{1}{\alpha^s}{}_{s+1}F_s\left(\begin{array}{c}
1,\alpha,\alpha,\alpha,\cdots\\
1+\alpha,1+\alpha,1+\alpha,\cdots\\
\end{array}\mid z\right).
</math>
Asymptotic expansion
The polylogarithm function <math>\mathrm{Li}_n(z)</math> is defined as
:<math>\mathrm{Li}_0(z)=\frac{z}{1-z}, \qquad \mathrm{Li}_{-n}(z)=z \frac{d}{dz} \mathrm{Li}_{1-n}(z).</math>
Let
:<math>
\Omega_{a} \equiv\begin{cases}
\mathbb{C}\setminus[1,\infty) & \text{if } \Re a > 0, \\
{z \in \mathbb{C}, |z|<1} & \text{if } \Re a \le 0.
\end{cases}
</math>
For <math>|\mathrm{Arg}(a)|<\pi, s \in \mathbb{C}</math> and <math>z \in \Omega_{a}</math>, an asymptotic expansion of <math>\Phi(z,s,a)</math> for large <math>a</math> and fixed <math>s</math> and <math>z</math> is given by
:<math>
\Phi(z,s,a) = \frac{1}{1-z} \frac{1}{a^{s
+
\sum_{n=1}^{N-1} \frac{(-1)^{n} \mathrm{Li}_{-n}(z)}{n!} \frac{(s)_{n{a^{n+s
+O(a^{-N-s})
</math>
for <math>N \in \mathbb{N}</math>, where <math>(s)_n = s (s+1)\cdots (s+n-1)</math> is the Pochhammer symbol.
Let
:<math>f(z,x,a) \equiv \frac{1-(z e^{-x})^{1-a{1-z e^{-x.</math>
Let <math>C_{n}(z,a)</math> be its Taylor coefficients at <math>x=0</math>. Then for fixed <math>N \in \mathbb{N}, \Re a > 1</math> and <math>\Re s > 0</math>,
:<math>
\Phi(z,s,a) - \frac{\mathrm{Li}_{s}(z)}{z^{a
=
\sum_{n=0}^{N-1}
C_{n}(z,a) \frac{(s)_{n{a^{n+s
+
O\left( (\Re a)^{1-N-s}+a z^{-\Re a} \right),
</math>
as <math>\Re a \to \infty</math>.
Software
The Lerch transcendent is implemented as LerchPhi in Maple and Mathematica, and as lerchphi in mpmath and SymPy.
References
- .
- . (See § 1.11, "The function Ψ(z,s,v)", p. 27)
- . (Includes various basic identities in the introduction.)
- .
- .
- .
External links
- .
- Ramunas Garunkstis, Home Page (2005) (Provides numerous references and preprints.)