Collection of Infinite Products and Series
Dr. Andreas Dieckmann, Physikalisches Institut der Uni Bonn
My interest in infinite products was started in the year 2000 by the problem of the electrical field of a line charge
trapped inside a rectangular box. After I learned that the double product can be solved with elliptic theta functions,
my collectors passion was triggered. The site has been growing ever since.
These pages deal with products, sums and other mathematical expressions in the following sections:
- Infinite Products
- Products involving Theta Functions
- Other formulae and curiosities including sums of hyperbolic and inverse tangent (arctan) functions and q - series
- q-Series
- special values of EllipticK and EllipticE
- Series of Hyperbolic Functions
- Series of CosIntegral
- some Limits
- diverse Series
- Series of Logarithms
- Series of Inverse Tangents ( Arcustangent )
- Series of Bessel Functions
- Series of Zeta PolyGamma PolyLog and related
- Series of Beta Functions
- Series of Gamma Functions
- a few Integrals
- iterated expressions ( Tetration )
- some properties of ProductLog LerchPhi and PolyLog
{j,n, m} are Integer; {λ, q} > 0 and r are real; { z, z1, z2, z3, z4} may be complex; Γ[a] is Gamma[a];
some of the products possess pointlike poles, where the denominator of a factor gets zero for certain
values of z. Some of the expressions are well known, others may be not; some were found in the
depths of the world wide web, the first are derived from the following product below.
- any formula you decide to use should be numerically tested for validity in the users domain -
Infinite Products : ( Back to Top )
This product converges and delivers infinite product representations for many functions if the {a,b,c,d} are
replaced by constants and simple polynoms in z :
Eulers product:
Products involving Theta Functions ( Back to Top )
The theta functions may be expressed through each other:
and exhibit a kind of double periodicity:
With m = InverseEllipticNomeQ[Exp[-π λ]] and K[m] = EllipticK[m]:
In the following is ( 0 < q < 1 ) and 
[ 0 , q ] , (
[ 0 , q ] =
[ 0 , -q ] ) :
m = InverseEllipticNomeQ[q] and K[m] = EllipticK[InverseEllipticNomeQ[q]]:
InverseEllipticNomeQ m[q] and K[m[q]] expressed through infinite products :
and 
can be expressed through m[q] , K[m[q]] and E[m[q]] :
and similarly :
and:
and from combining the above like:
we get:
If the result of the imaginary transformation doesn't seem right, consider the following points:
• If in the resulting formula a sign change of the imaginary part as function of q occurs under a square root ( at q = Exp[- π / 2] ) then the square root may take the other sign
• Logs with complex arguments may end up on a wrong branch, try replacing Log[...] with Log[...] + n I 2 π
The condition q ≤ Exp[-Pi / 2] is hopefully temporary caused by an error of Mathematica in evaluating elliptic Integrals at certain complex arguments...
Theta Functions (z = 
) expressed through EllipticK and m :
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Series expansion of InverseEllipticNomeQ:
Series expansion of EllipticNomeQ:
specific values:
Theta Functions , specific values :
The missing ϑ functions we get from
Other formulae and curiosities including sums of hyperbolic and inverse tangent (arctan) functions and q - series: ( Back to Top )
q - Series : ( Back to Top )
with 
→ Cosh[ k Log[ q ]] + Sinh[ k Log[ q ]] the following expressions can be transformed into sums of hyperbolic functions.
The inner sum above gives the number of ascending sequences of length k in the permutations of n numbers.
For integer n PolyLog[-n, q] appears as a rational function in q.
( m = InverseEllipticNomeQ[q], K[m] = EllipticK[m], E[m] = EllipticE[m] ):
The introduction of QPolyGamma[n, z, q] (nth derivative of QDigamma function (z, q)) in Mathematica 7 allows expression of
qHypergeometric2F1[a,b,c,q,x] is a q-variant of 2F1:
special values of EllipticK and EllipticE: ( Back to Top )
E[m] is EllipticE[m];
Series of Hyperbolic Functions: ( Back to Top )
m = InverseEllipticNomeQ[
] :
Series of CosIntegral: ( Back to Top )
some Limits : ( Back to Top )
diverse Series : ( Back to Top )
The sum 
gives following results for some rational s :
This sum alternates between ± π for z ∈ N :
In the following 4 expressions b =
:
The next two expressions contain s = 
and t = 
:
Series of Logarithms : ( Back to Top )
( m = InverseEllipticNomeQ[q], K[m] = EllipticK[m], E[m] = EllipticE[m] ):
Series of Inverse Tangents ( Arcustangent ) : ( Back to Top )
( m = InverseEllipticNomeQ[q], K[m] = EllipticK[m], E[m] = EllipticE[m] ):
If an expression contains at the end ' + n π , (**) ', that means it has discontinuities in the Log , so you may have to add multiples of π to get correct results…
Series of Bessel Functions : ( Back to Top )
Series of Zeta, PolyGamma, PolyLog and related : ( Back to Top )
Series of Beta Functions : ( Back to Top )
Recurrence relation : Beta[x, a + 1, b] + Beta[x, a, b + 1] = Beta[x, a, b];
Series of Gamma Functions : ( Back to Top )
( K[x] = EllipticK[x] ):
a few Integrals : ( Back to Top )
Substitute 
and the Feynman - Hibbs Integral
iterated expressions ( Tetration ) : ( Back to Top )
The above function f[x] = - ProductLog[-Log[x]] / Log[x] has a special 'swapping' symmetry of basis and exponent in its argument: 
.
f[x] is not defined beyond the maximum of its inverse function 
, namely 
< x , so with this symmetry it is plausible that the exponential tower
doesn't converge for x < 
as well , where it shows a bifurcation.
some properties of ProductLog, LerchPhi and PolyLog ( Back to Top )
For 1/e ≤ x is ProductLog[ Log[x] x] = Log[x] .
For 0 ≤ x ≤ e is ProductLog[-Log[x]/x] = -Log[x] .
For 0 ≤ x is Log[ProductLog[x]] = Log[x] - ProductLog[x] .
For purely imaginary arguments (x ∈ R) the complex decomposition of LerchPhi is:
These carry over with a = 0 to PolyLog:
The imaginary part of LerchPhi[x , s , a] with 1 ≤ x ∈ R is given by :
And with a = 0 follows the imaginary part of PolyLog[ s , x] :
The complex decomposition of 
with 1 ≤ x ∈ R and 0 ≤ {b, s} ∈ N into real and imaginary part can be obtained by the following expression:
explicitly for low s and b = 2 :
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For all z ∈ C not on the real axis in ( - ∞ < z < 1) and 0 ≤ {b, s} ∈ N the following inversion identity holds
(the If statement makes a '+' in case of an imaginary part of z greater than zero, a '-' in all other cases):
The real part of 
with 1 ≤ x ∈ R is also given by
For (b ∈ N) is
The real and imaginary parts of LerchPhi[ 
, 2, 1/2 ] (on the unit circle) are
With Clausen type functions for LerchPhi defined as
(0 < s ∈ Integer, 0 ≤ θ ≤ 2π, the even CLi and the odd SLi are expressible through Euler Polynomials),
the real and imaginary parts of 
(on the unit circle) are
the expressions for 
with lowest s being
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The above polynomials in a make nice approximations to trigonometric functions, getting better with increasing n.
The first non polynomial partnerfunctions are found to be
The function 
has an interesting derivative :
that means the lower CLi and SLi are essentially derivatives of the higher ones.
With Clausen type functions defined as
(0 < s ∈ Integer, 0 ≤ θ ≤ 2π, the even Ci and the odd Si are expressible through Bernoulli Polynomials),
the real and imaginary parts of 
(on the unit circle) are
the expressions for 
with lowest s being
The above polynomials in a make nice approximations to trigonometric functions, getting better with increasing s:
As before the derivative of 
is 
with lowered index.
The first non polynomial partnerfunctions are found to be
The complex decomposition of PolyLog[s, x] with 1 ≤ x ∈ R and 0 ≤ s ∈ N can be obtained by the following expression:
explicitly for low s :
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For all z ∈ C and not on the real axis in ( 0 ≤ z < 1 ) and 0 ≤ {b, s} ∈ N the following inversion identity holds :
LerchPhi and PolyLog display a similar (alternating with s) scheme in their real and imaginary parts :
The lowest Bernoulli and Euler Polynomials are
| BernoulliB | EulerE | |
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