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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,
|z|<1; s>1,|z|=1.

If s is not an integer then |pha|<π; if s is a positive integer then a0,-1,-2,; if s is a non-positive integer then a can be any complex number. 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).