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25 Zeta and Related FunctionsRelated Functions

§25.14 Lerch’s Transcendent


§25.14(i) Definition

25.14.1 Φ(z,s,a)=n=0zn(a+n)s,
a0,-1,-2,,|z|<1; s>1,|z|=1.

For other values of z, Φ(z,s,a) is defined by analytic continuation. This is the notation used in Erdélyi et al. (1953a, p. 27). Lerch (1887) used 𝔎(a,x,s)=Φ(e2πix,s,a).

The Hurwitz zeta function ζ(s,a)25.11) and the polylogarithm Lis(z)25.12(ii)) are special cases:

25.14.2 ζ(s,a)=Φ(1,s,a),
s>1, a0,-1,-2,,
25.14.3 Lis(z)=zΦ(z,s,1),
s>1, |z|1.

§25.14(ii) Properties

With the conditions of (25.14.1) and m=1,2,3,,

25.14.4 Φ(z,s,a)=zmΦ(z,s,a+m)+n=0m-1zn(a+n)s.
25.14.5 Φ(z,s,a)=1Γ(s)0xs-1e-ax1-ze-xdx,
s>0, a>0, z[1,).
25.14.6 Φ(z,s,a)=12a-s+0zx(a+x)sdx-20sin(xlnz-sarctan(x/a))(a2+x2)s/2(e2πx-1)dx,
s>0 if |z|<1; s>1 if |z|=1,a>0.

For these and further properties see Erdélyi et al. (1953a, pp. 27–31).