Lerch Transcendent

Author

Eric W. Weisstein
October 22, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/LerchTranscendent.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LerchTranscendent.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Definition

(∑_(k=0)^∞)((z^k)/(((a+k))^s))==Φ(z,s,a)

v5.2

Sum[((z^k)/(((a+k))^s)),{k,0,∞}]

v6

Sum[((z^k)/(((a+k))^s)),{k,0,∞}]

v7

Sum[((z^k)/(((a+k))^s)),{k,0,∞}]

(∑_(k=0)^∞)((z^k)/((((((a+k))^2)))^(s/2)))==(LerchPhi[z, s, a])

Sum[((z^k)/((((((a+k))^2)))^(s/2))),{k,0,∞}]

(∑_(n=0)^∞)(((-(z/(1-z))))^n) (∑_(k=0)^n)(((-1))^k) (({{n}, {k}})) (((a+k))^(-s))==(1-z) Φ(z,s,a)

v5.2

Sum[(((((-z)/(1-z))))^n)Sum[(((-1))^k)Binomial[n,k](((a+k))^(-s)),{k,0,n}],{n,0,∞}]

v6

Sum[(((((-z)/(1-z))))^n)Sum[(((-1))^k)Binomial[n,k](((a+k))^(-s)),{k,0,n}],{n,0,∞}]

v7

Sum[(((((-z)/(1-z))))^n)Sum[(((-1))^k)Binomial[n,k](((a+k))^(-s)),{k,0,n}],{n,0,∞}]

Special Cases

Φ(z,n,1)==(((Subscript[Li, n])(z))/z)

LerchPhi[z,n,1]

V7

LerchPhi[z,n,1]

HurwitzLerchPhi[z,n,1]

β(s)==(∑_(k=0)^∞)(((-1))^k) (((2 k+1))^(-s))

Sum

v5.1

Sum[(((-1))^k)(((2k+1))^(-s)),{k,0,∞}]//FullSimplify

v5.2

Sum[(((-1))^k)(((2k+1))^(-s)),{k,0,∞}]//FullSimplify

FunctionExpand

Table[(2^(-s))LerchPhi[-1,s,(1/2)],{s,2,5}]//FullSimplify

v5.1

(2^(-s))FunctionExpand[LerchPhi[-1,s,(1/2)]]

v5.2

(2^(-s))FunctionExpand[LerchPhi[-1,s,(1/2)]]

v6

(2^(-s))FunctionExpand[LerchPhi[-1,s,(1/2)]]//Simplify

V7

Sum[(((-1))^k)(((2k+1))^(-s)),{k,0,∞}]

FullSimplify[%]

(2^(-s))FunctionExpand[LerchPhi[-1,s,(1/2)]]//FullSimplify

(2^(-s))FunctionExpand[HurwitzLerchPhi[-1,s,(1/2)]]//FullSimplify

Special Values

(1/4) Φ(-1,2,(1/2))==C

V6

(1/4)LerchPhi[-1,2,(1/2)]

V7

(1/4)LerchPhi[-1,2,(1/2)]

(1/4)HurwitzLerchPhi[-1,2,(1/2)]

Derivative[0,1,0][LerchPhi](-1,-1,1)==log(((A^3)/((2^(1/3)) (e^(1/4)))))

V5

Derivative[0, 1, 0][LerchPhi][-1, -1,1]

V7

Derivative[0, 1, 0][LerchPhi][-1, -1,1]

Derivative[0, 1, 0][HurwitzLerchPhi][-1, -1,1]

Derivative[0,1,0][LerchPhi](-1,-2,1)==((7 ζ(3))/(4 (π^2)))

V5

Derivative[0, 1, 0][LerchPhi][-1, -2, 1]

V7

Derivative[0, 1, 0][LerchPhi][-1, -2, 1]

Derivative[0, 1, 0][HurwitzLerchPhi][-1, -2, 1]

Derivative[0,1,0][LerchPhi](-1,-1,(1/2))==(C/π)

V5

Derivative[0, 1, 0][LerchPhi][-1, -1, 1/2]

V7

Derivative[0, 1, 0][LerchPhi][-1, -1, 1/2]

Derivative[0, 1, 0][HurwitzLerchPhi][-1, -1, 1/2]

(1/2) Derivative[0,1,0][LerchPhi]((1/2),0,1)==Derivative[1,0][PolyLog](0,(1/2))

V5

(1/2)*Derivative[0, 1, 0][LerchPhi][1/2, 0, 1]

V7

(1/2)*Derivative[0, 1, 0][LerchPhi][1/2, 0, 1]

(1/2)*Derivative[0, 1, 0][HurwitzLerchPhi][1/2, 0, 1]

Derivative[0,1,0][LerchPhi](-1,1,(1/2))==(1/2) π (log(((8 (π^3))/((Γ((1/4)))^4)))+ℽ)

V5

Derivative[0, 1, 0][LerchPhi][-1, 1, 1/2]

V7

Derivative[0, 1, 0][LerchPhi][-1, 1, 1/2]//FullSimplify

Derivative[0, 1, 0][HurwitzLerchPhi][-1, 1, 1/2]//FullSimplify

Integrals

(∫_0^∞)((k^s)/((e^(k-μ))+1))dk==-s (Gamma[s]) (PolyLog[s+1, -(e^μ)])

Fermi-Dirac Integral

V5

Integrate[((k^s)/(Exp[k-μ]+1)),{k,0,∞},Assumptions->Re[s]>-1]

FullSimplify[%]

V7

Integrate[((k^s)/(Exp[k-μ]+1)),{k,0,∞},Assumptions->Re[s]>-1]

FullSimplify[%]

(∫_0^∞)((k^s)/((e^(k-μ))-1))dk==(Gamma[s+1]) (PolyLog[s+1, (e^μ)])

Bose-Einstein Integral

V5

Integrate[((k^s)/(Exp[k-μ]-1)),{k,0,∞},Assumptions->Re[s]>-1]

FullSimplify[%]

V7

Integrate[((k^s)/(Exp[k-μ]-1)),{k,0,∞},Assumptions->Re[s]>-1]

FullSimplify[%]

(1/(Γ(s)))(∫_0^∞)(((e^(-(u-1) t)) (t^(s-1)))/((e^t)-z))dt

Guillera and Sondow, Lemma 2.1

V6

(1/(Gamma[s]))Integrate[((Exp[-(u-1)t])/(Exp[t]-z))(t^(s-1)),{t,0,∞}]

V7

(1/(Gamma[s]))Integrate[((Exp[-(u-1)t])/(Exp[t]-z))(t^(s-1)),{t,0,∞}]

(∫_0^∞)((LerchPhi[-z, s, u])/(z+1))dz

Integrate[((LerchPhi[-z,s,u])/(1+z)),{z,0,∞}]

With[{s=1,u=2},{Integrate[((LerchPhi[-z,s,u])/(1+z)),{z,0,∞}],s LerchPhi[1,s+1,u]}]

(∫_0^1)(∫_0^1)(((((-ln(x y)))^s) ((x^(u-1)) (y^(v-1))))/(1-x y z))dydx

Integrate[(((x^(u-1))(y^(v-1)))/(1-x y z))(((-Log[x y]))^s),{x,0,1},{y,0,1}]//Timing

Integrate[(((x^(u-1))(y^(v-1)))/(1-x y z))(((-Log[x y]))^s),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)(((((x y))^(u-1)) (((-ln(x y)))^s))/(1-x y z))dydx

Integrate[((((x y))^(u-1))/(1-x y z))(((-Log[x y]))^s),{x,0,1},{y,0,1}]//Timing

Integrate[((((x y))^(u-1))/(1-x y z))(((-Log[x y]))^s),{y,0,1},{x,0,1}]//Timing