A Generalization of Fourier Series occurring in Atomic Theory

TL;DR

This paper generalizes certain Fourier series used in atomic theory, specifically in the semi-classical model, expanding their mathematical framework and potential applications.

Contribution

It introduces a broader class of Fourier series relevant to atomic physics, extending previous specific cases in semi-classical atomic models.

Findings

Generalized Fourier series applicable to atomic models

Enhanced mathematical tools for semi-classical atomic theory

Potential for improved analytical techniques in physics

Abstract

A number of the Fourier Series which occur in the theory of the semi-classical atom due to Englert and Schwinger are generalized and presented.

Equations

$$\{z\}=frac(z)=z-\lfloor z\rfloor ,0\le \{z\}<1,$$

$$<z>=z-\lfloor z+1/2\rfloor ,-\frac{1}{2}\le <z><\frac{1}{2},$$

$$\{z\}=<z-\frac{1}{2}>+\frac{1}{2}.$$

$$\sum _{k=1}^{\infty }\frac{(-1{)}^k\sin (2\pi kz)}{\pi k}=-<z>,$$

$$S_0(z)$$

$$C_1(z)$$

$$S_1(z)$$

$$C_2(z)$$

$$S_2(z)$$

$$C_3(z)$$

$${\int }_0^z$$

$${\mathcal{S}}_n(z)$$

$${\mathcal{C}}_n(z)$$

$${\mathcal{S}}_n(z)$$

$${\mathcal{C}}_{n+1}(z)$$

$${\mathcal{S}}_0(z)$$

$$C_1(z)$$

$${\mathcal{C}}_n(z)=\sum _{i=0}^n{c}_i(n)<z{>}^{2i},$$

$${\mathcal{C}}_n(z)=-4{\int }_0^z{\int }_0^{\zeta }{\mathcal{C}}_{n-1}({\zeta }^{\prime })d{\zeta }^{\prime }d\zeta -\frac{(2^{2n+1}-1)}{(2n+2)!}|B_{2n+2}|,$$

$$c_0(n)$$

$$c_1(n)$$

$$c_i(n)$$

$$\sum _{i=0}^{n-1}\frac{c_i(n-1)}{2^{2i}(2i+1)}=0.$$

$${\mathcal{C}}_n(z)=\left[\sum _{i=0}^{n-1}\frac{c_i(n-1)}{2^{2i-1}(2i+1)}\right]<z{>}^2+2\sum _{i=2}^n\frac{c_{i-1}(n-1)}{i(2i-1)}<z{>}^{2i}-\frac{(2^{2n+1}-1)}{(2n+2)!}|B_{2n+2}|.$$

$${\mathcal{S}}_n(z)=2\sum _{i=1}^{n-1}\frac{c_i(n)}{(2i+1)}[<z{>}^{2i}-\frac{1}{4^i}].$$

$${\mathcal{S}}_n^{\prime }(z)$$

$${\mathcal{C}}_n^{\prime }(z)$$

$${\mathcal{S}}_0^{\prime }(z)=-\frac{1}{2}\tan (\pi z),$$

$${\mathcal{C}}_0^{\prime }(z)=-\frac{1}{\pi }\ln |2\cos (\pi z)|,$$

$${\mathcal{S}}_n^{\prime }(z)$$

$${\mathcal{C}}_n^{\prime }(z)$$

$$L{i}_a(z)=\sum _{k=1}^{\infty }\frac{z^k}{k^a},$$

$$L{i}_1(z)=-\ln (1-z).$$

$${\tilde{\mathcal{S}}}_0(z)=\sum _{k=1}^{\infty }\frac{\sin (2\pi kz)}{\pi k}=\frac{1}{2}-\{z\}=-<z-1/2>,$$

$${\tilde{\mathcal{C}}}_1(z)=\sum _{k=1}^{\infty }\frac{\cos (2\pi kz)}{(\pi k{)}^2}=<z-1/2{>}^2-\frac{1}{12},$$

$${\tilde{\mathcal{S}}}_n(z)$$

Full text

1 Some Generalized Fourier Series

In the work shown below we have set the goal of presenting certain Fourier series in terms of closed form expressions which contain the special functions commonly found in symbolic computational software such as Maple and Mathematica. In advance of the derivation of the sums to be obtained below it will prove useful to introduce the functions $\{z\}$ and $<z>$ i.e.

[TABLE]

[TABLE]

where $\{z\}$ is the fractional part of $z$ and $\lfloor z\rfloor$ is the floor function. We note that

[TABLE]

1.1 Englert alternating sums

The alternating sum

[TABLE]

has been given by Englert, et al. [1].  Using that expression, the sums given below have been obtained by repeated integration of that series or obtained from the literature [2], [3] and represent an extension of Englert’s work.  We have in the first instances sums which are expressible as polynomials which contain the function $<z>$ i.e.

[TABLE]

General expressions for these sum can be obtained and are presented in the sequel.  Important in what follows we note the integral relations

[TABLE]

Defining the sums $\mathcal{S}_{n}(z)$ and $\mathcal{C}_{n}(z)$ whose summands are even functions of the index k\by

[TABLE]

we get on integration of $\mathcal{C}_{n}(z)$ and $\mathcal{S}_{n}(z)$

[TABLE]

where the $B_{r}$ are the Bernoulli numbers [4].  Using expressions for the initial quantities i.e.

[TABLE]

all of the higher sums can step-by-step be computed.

It is also possible to obtain general expressions for these sums.  If  we write

[TABLE]

together with the use of (3) and (4) to obtain the ‘recursion’ relation

[TABLE]

we can show that the coefficients $c_{i}(n)$ contained in the polynomial expressions for $\mathcal{C}_{n}(z)$ are related to the coefficients $c_{i}(n-1)$ of the previous polynomial $\mathcal{C}_{n-1}(z)$.  We get

[TABLE]

and

[TABLE]

Finally we have

[TABLE]

In a similar way we also have

[TABLE]

In the second instance, the remaining alternating sums  which are odd functions of the index $k$ will be seen to contain expressions which are more complicated .  Writing the sums as

[TABLE]

we have for $\mathcal{S}_{0}^{\prime}(z)$ [5]

[TABLE]

and for $\mathcal{C}_{0}^{\prime}(z)$ [6]

[TABLE]

then (see the Appendix)

[TABLE]

where $Li_{a}(z)$ is the polylogarithm function of order $a$ [7]

[TABLE]

[TABLE]

The latter quantities being one of the standard functions contained in Maple and Mathematica.

1.2 Non-alternating Englert sums

The non-alternating sum

[TABLE]

is due to Titchmarch [8].  Then

[TABLE]

More generally we define the families of non-alternating even and odd sums by

[TABLE]

with  [9]

[TABLE]

and

[TABLE]

respectively.  We also note the general relations of these sums to the corresponding alternating even and *odd *sums given above i.e.

[TABLE]

Those relations follow directly from the definitions of the ‘hatted’ functions or the integrated expression of $\widetilde{\mathcal{S}}_{n}(z)$ and $\widetilde{\mathcal{C}}_{n}(z).$ Here we get

[TABLE]

[TABLE]

In this instance we note the integral relations

[TABLE]

Using (6), (7) the sums listed below are seen to be polynomials in the functions $<z-1/2>$.  That is to say, the non-alternating sums are just the sums $\mathcal{S}_{n}(z),$ $\mathcal{C}_{n}(z)$ and $\mathcal{S}_{n}^{\prime}(z),$ $\mathcal{C}_{n}^{\prime}(z)$ which contain the shifted variable i.e. $z\rightarrow z-1/2.$  We have

[TABLE]

and

[TABLE]

1.3 Series with summands containing arguments $2k+1$

We write the alternating and non-alternating sums which are even and odd functions in the argument $(2k+1)$ as

[TABLE]

and

[TABLE]

respectively.

We find that the alternating and non-alternating sums are interrelated i.e.

[TABLE]

As seen above repeated integrations produce the higher sums. The first few of the even alternating sums being (cf. Appendix)

[TABLE]

and for the odd alternating sums

[TABLE]

The first few of the non-alternating sums can then be obtained from (8).

1.4 Modified Englert sums

Here we consider sums in which the arguments of the trigonometric functions in the summands are $2\pi kz$ whereas the arguments in the denominator are $(2k+1)$ i.e.

[TABLE]

and

[TABLE]

From (9), and (10) above we get

[TABLE]

We see that the sums above are not new in that they are related to the sums $\mathbf{S}_{n}(z),$ $\mathbf{C}_{n}(z),$ $\mathbf{S}_{n}^{\prime}(z),$and $\mathbf{C}_{n}^{\prime}(z)$ together with $\ \widetilde{\mathbf{S}}(z),\widetilde{\mathbf{C}}_{n}(z),\widetilde{\mathbf{S}}_{n}^{\prime}(z),$ and $\widetilde{\mathbf{C}}_{n}^{\prime}(z)$**.  **The first few of these sums are

[TABLE]

and

[TABLE]

Appendix

In this appendix we show that in special cases the polylogarith functions found in the relations above can be written as polynomials in the variables $<f(z)>$.  We have

[TABLE]

from which it follows that

[TABLE]

In a similar way we have

[TABLE]

In the cases where $z$ has been replaced by $z\pm 1/4$ the relations are more complicated i.e.

[TABLE]

The functions $f_{i,n}$ are used here to indicate that these Real and Imaginary parts of the $Li_{a}$ functions can be expressed in terms of polynomials in $<F(z)>$  where $F(z)$ is some linear function of $z$.  In the remaining equations above, this does not appear to be possible even in the cases of infinite series in $<F(z)>$.