_{1}

^{*}

This paper presents closed-form expressions for the series,
, where the sum is from
*n* = 1 to
*n* = ∞. These expressions were obtained by recasting the series in a different form, followed by the use of certain relationships involving the elliptical nome. Among the values of
*x* for which these expressions can be obtained are of the form:
and
, where l is an integer between
–∞ and ∞. The values of
*λ* include 1,
,
and 3. Examples of closed-form expressions obtained in this manner are first presented for
,
,
, and
. Additional examples are then presented for
,
,
, and
. This undertaking was prompted by the author’s work on an electrostatics boundary-value problem related to the van der Pauw measurement technique of electrical resistivity. The presence of this series for
*x* =
in the solution of that problem and its absence from any compendium of infinite series that he consulted led to this work.

The series S(x), where

does not appear in any of several compendiums of infinite series or products [

The particular measurement configuration considered here is shown in

Depicted in this figure there is a square sample of side “a” and a square contact array of semi-diameter w, rotated by 45˚ with respect to the sample and displaced from the center of the sample by the vector s. This displacement can be in any direction and magnitude as long as the array remains completely within the sample (including on the boundary). Current I_{o} is forced to enter contact 1 and leave contact 2, and the resulting potential difference between contacts 4 and 3, ∆φ = φ_{4} − φ_{3}, is determined from a voltage measurement. Mathematically, the contacts are considered point-like, which in practice means that their diameter

This is a special case of the more general one considered by van der Pauw [

In this equation, ρ is the electrical resistivity. When w/a < 1/2 and s ≠ 0, Equation (2) must be modified, such that:

Clearly, F(1/2, 0) = ln(2). The function F(w/a, s) was calculated in [

Allowing w/a = 1/2, we obtain, after considering even and odd n separately:

Using the relationship [

and F(1/2, 0) = ln(2), we obtain:

It is clear that the sum in Equation (7) should be very close to ln(2) because coth(nπ) → 1 so quickly with n. In fact, the right-hand side of Equation (7) is 0.6968×××, while ln(2) = 0.6931×××, resulting in a percentage difference of about 0.5%. When the left-hand side was summed directly, the two sides agreed to the limit of precision of the software used, or 15 significant figures.

The above derivation is based on a requirement of a physics problem and applies to only one value of x, x = π. The following derivation, which applies to an array of values of x, proceeds by recasting the series in a different form and applying certain existing relationships to evaluate the resulting series.

Using the definition of the hyperbolic cotangent, we rewrite S(x) as:

This series can be expressed in a more useful form for our current purpose. We first note that for

For certain values of x, the series can now be evaluated in a straightforward manner, using existing relationships.

These relationships are stated in [

In this expression, q is the elliptical nome [

Second, in [

Using this equation to form the right-hand side of Equation (11) easily leads to its left-hand side, after a few simple algebraic manipulations. The equality between the first and second terms in Equation (12) is discussed in [

Also in the second section of [

or its inverse,

and

The first of these three equations allows one to generate the inverse nome of q^{2}, given the inverse nome of q, and that of q^{4} from that of q^{2}, etc. From Equation (14), the inverse nome of q^{1/2}, q^{1/4}, etc. can be generated starting with that of q. Thus, starting with a particular value of X, say X_{0}, one can generate an entire array of X, say X_{n}_{}, to which Equation (10) can be applied, such that:

By using Equation (15), which follows directly from the definition of elliptical nome in [_{0} through the inverse of λ (i.e., X_{0} = πλ/2 → X_{0} = π/2λ). This has been done for a few cases in [

A few examples of the use of these equations to generate closed-form expressions for S(x) will now be presented. They will be for λ = 1/2, 1, 2, and 4, which corresponds to x = π/4, π/2, π, and 2π, respectively. Since the value of S(x) for x = π motivated this entire undertaking, it is included here. Starting with λ = 1, for which m = 1/2, one can use Equation (14) to generate the value of m for λ = 1/2 and Equation (16) to generate m for λ = 2 and 4. The value of m for λ = 1/2 can also be generated from the value of m for λ = 2, using Equation (15). All of these results are actually presented in [

All of these results (and others not shown) have been confirmed by an agreement with a direct summation of each series.

Simple analytic formulas have been produced for eight values of x for the series

The author is grateful to J. Mercer of Albuquerque, New Mexico for reviewing the manuscript and making valuable suggestions for its improvement. This work was self-funded.