the application of real convolution for analytically...

Research ArticleThe Application of Real Convolution for Analytically EvaluatingFermi-Dirac-Type and Bose-Einstein-Type Integrals

Jerry P Selvaggi 1 and Jerry A Selvaggi2

1Private Practice Schenectady NY 12308 USA2Private Practice Pittsburgh PA 15217 USA

Correspondence should be addressed to Jerry P Selvaggi jpsst44gmailcom

Received 13 September 2017 Revised 6 March 2018 Accepted 20 March 2018 Published 6 May 2018

Academic Editor N K Govil

Copyright copy 2018 Jerry P Selvaggi and Jerry A Selvaggi This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The Fermi-Dirac-type or Bose-Einstein-type integrals can be transformed into two convergent real-convolution integrals Thetransformation simplifies the integration process andmay ultimately produce a complete analytical solutionwithout recourse to anymathematical approximations The real-convolution integrals can either be directly integrated or be transformed into the LaplaceTransform inversion integral in which case the full power of contour integration becomes available Which method is employed isdependent upon the complexity of the real-convolution integral A number of examples are introduced which will illustrate theefficacy of the analytical approach

1 Introduction

This article presents an extension to themethod developed byJ A Selvaggi and J P Selvaggi [1] for analytically evaluatingFermi-Dirac-type andBose-Einstein-type integralsTheFermi-Dirac and Bose-Einstein integrals occupy an important rolein areas such as solid-state physics and statistical mechanicsThere exists numerous approximate analytical methods forevaluating these integrals but none of the publishedmethodsgive a general technique for analytically evaluating theseintegrals valid for the full range of the degeneracy parameterThis article illustrates a general method for attacking theseintegrals using real convolution

The right-hand side of (1) and (2) will be defined asFermi-Dirac-type integral and the Bose-Einstein-type integralrespectively

119865 (120585) = intinfin0

119891 (119909)1 + 119890119909minus120585 119889119909 (1)

119861 (120585) = intinfin0

119891 (119909)minus1 + 119890119909minus120585 119889119909 (2)

Analytical evaluation of these integrals yields the functions119865(120585) and 119861(120585) These functions will be defined as the Fermi-Dirac function and the Bose-Einstein function respectivelyThe parameter 120585 is called the degeneracy parameter a termencountered in statistical mechanics However as far as theintegrals in (1) and (2) are concerned 120585 represents any realnumber In general integrals of this type do not allow forclosed-form solutions in terms of elementary functions Thisarticle introduces a generalmethod for analytically evaluatingthe integrals given in (1) and (2) for various functions119891(119909) The denominator of the integrands in (1) or (2) isexactly that found in the familiar Fermi-Dirac [2ndash4] or Bose-Einstein integrals [5 6]These integrals are often encounteredin statistical and quantum statistical mechanics [7ndash9] Theauthors will mainly consider the Fermi-Dirac functions andBose-Einstein functions within that domain for which 120585 isinR ge 0 If 120585 isin R lt 0 (1) and (2) may be solved by elementarymethods

Numerous techniques have been employed to analyticallyapproximate and numerically evaluate the half-order Fermi-Dirac functions [2 10ndash18] and half-order Bose-Einstein func-tions [5 17 18] (see Appendix) A relatively new representation

HindawiJournal of Complex AnalysisVolume 2018 Article ID 5941485 8 pageshttpsdoiorg10115520185941485

2 Journal of Complex Analysis

for these integrals for 119891(119909) = 119909120592 forall120592 gt minus1 is the Poly-logarithm function [19ndash22] This function has been studiedextensively in the literature In fact mathematical softwaresuch as Mathematica [23] uses a Polylogarithm algorithmto numerically compute the Fermi-Dirac and Bose-Einsteinintegrals However the authors will illustrate that the appli-cation of real convolution allows for the complete analyticalevaluation of the integrals in (1) and (2) for a wide rangeof functions 119891(119909) The authors have already employed thistechnique [1] to analytically evaluate the integral in (1) forthe well-known and important case of the half-order Fermi-Dirac functions where 119891(119909) = 119909119898minus12 forall119898 isin Zge A fewexamples will be considered which will help to illustratethe efficacy of the method Each solution was numericallychecked by employingMathematica [23] and other numericalalgorithms

2 Theoretical Development

Rewrite the integrals given in (1) and (2) in terms of twoconvergent real-convolution integrals [24] The observationthat real convolution can be employed is the main focus ofthis article In fact once the integrals are transformed intothe proper real-convolution form the difficulty in analyticallyevaluating (1) and (2) may be substantially reduced

In order to see that (1) and (2) can each be transformedinto two convergent real-convolution integrals requires thateach integral be put into the proper form To this end rewrite(1) and (2) as follows

119865 (120585) = int1205850

119891 (119909)1 + 119890minus(120585minus119909) 119889119909 + intinfin

120585

119891 (119909) 119890minus(119909minus120585)1 + 119890minus(119909minus120585) 119889119909 (3)

119861 (120585) = minusint1205850

119891 (119909)1 minus 119890minus(120585minus119909) 119889119909 + intinfin

120585

119891 (119909) 119890minus(119909minus120585)1 minus 119890minus(119909minus120585) 119889119909 (4)

valid forall120585 isin R ge 0The denominators in each of the integrals in (3) and (4)

can be expanded in a binomial expansion Each expansionresults in a convergent integral within its limits of integrationEmploying (3) and expanding the denominator of bothintegrals in their appropriate binomial expansion yields

119865 (120585) = int1205850119891 (119909) 119889119909 + infinsum

119901=1

(minus1)119901

sdot [int1205850119891 (119909) 119890minus119901(120585minus119909)119889119909 minus intinfin

120585119891 (119909) 119890minus119901(119909minus120585)119889119909]

(5)

Likewise (4) can be rewritten as

119861 (120585) = minusint1205850119891 (119909) 119889119909

minus infinsum119901=1

[int1205850119891 (119909) 119890minus119901(120585minus119909)119889119909 minus intinfin

120585119891 (119909) 119890minus119901(119909minus120585)119889119909]

(6)

Depending upon the complexity of the function 119891(119909)direct integration may be possible However the authors will

develop the necessary mathematical machinery employingthe Laplace Transform inversion integral and contour inte-gration of complex variable theory in order to analyticallyevaluate (5) and (6) The two integrals within the summationin (5) for example are real-convolution integrals [24] definedas follows

int1205850119891 (119909) 119890minus119901(120585minus119909)119889119909 = 12120587119894 int

120590+119894infin

120590minus119894infin

F (119904) 119890120585119904(119904 + 119901) 119889119904intinfin120585

119891 (119909) 119890minus119901(119909minus120585)119889119909 = 12120587119894 int120590+119894infin


F (119904) 119890120585119904(minus119904 + 119901)119889119904(7)

where F(119904) is defined as the Laplace Transform of 119891(119909) Bothexpressions are valid forforall120585 isin R gt 0One can eventually relaxthe restriction on 120585 to include forall120585 isin R ge 0 Substituting (7)into (5) yields the following

119865 (120585) = int1205850119891 (119909) 119889119909

+ 2 infinsum119901=1

(minus1)119901 12120587119894 int120590+119894infin


119904F (119904) 119890120585119904(1199042 minus 1199012)119889119904(8)

Equation (8) represents an exact alternative expression for(1) The only restriction put upon 119891(119909) is that it must be afunction which allows the integral in (1) to be convergentOf course 119891(119909) must have a Laplace Transform The sameprocedure illustrated above yields the following expressionfor the Bose-Einstein-type integral

119861 (120585) = minusint1205850119891 (119909) 119889119909 minus 2 infinsum

119901=1

12120587119894 int120590+119894infin


119904F (119904) 119890120585119904(1199042 minus 1199012)119889119904 (9)

The first integral shown in (8) or (9) may be quite simplefor practical problems if 119891(119909) is an analytically integrablefunction The second integral shown in (8) or (9) is theinverse Laplace Transform [24ndash26] defined as follows

119892 (120585 119901) = Lminus1 [119866 (119904)] = 12120587119894 int

120590+119894infin


119904F (119904) 119890120585119904(1199042 minus 1199012)119889119904 (10a)

where

119866 (119904) = 119904F (119904)(1199042 minus 1199012) (10b)

The contour defined by the limits of integration in (10a)is called the Bromwich Contour [24ndash26] Of course there isno guarantee that the analytical evaluation of the inversionintegral of (10a) will be an easy task However for manypractical problems found in the literature this appears not tobe a problem

3 Application of Real Convolution

31 Example 1 This simple example is used to verify themethod developed in Section 2 Let 119891(119909) = 119909119898 forall119898 isin Zge

Journal of Complex Analysis 3

and evaluate the expression for the Fermi-Dirac function byemploying (8) The result is as follows

119865119898 (120585)= 120585119898+1119898 + 1

+ 2 (119898) infinsum119901=1

(minus1)119901 12120587119894 int120590+119894infin


119890120585119904119904119898 (1199042 minus 1199012)119889119904(E11)

Evaluate the inversion integral in (E11) by applyingthe theory of residues of complex variable theory This isaccomplished by choosing the appropriate contour in thecomplex plane Figure 1 represents just one possible contourchosen to evaluate the integral in (E11) The contour isclosed in the left-hand plane in order to ensure convergence

Employing the contour of Figure 1 yields the following

12120587119894 119890120585119904119904119898 (1199042 minus 1199012)119889119904= 12120587119894 int

120590+119894infin


119890120585119904119904119898 (1199042 minus 1199012)119889119904= 119890120585119904119904119898 (119904 minus 119901)

100381610038161003816100381610038161003816100381610038161003816119904=119901119890120587119894+ 1(119898 minus 1) 119889

119898minus1

119889119904119898minus1 ( 1198901205851199041199042 minus 1199012)100381610038161003816100381610038161003816100381610038161003816119904=0

= minus 119890minus1205851199012119901119898+1119890119898120587119894 + 1(119898 minus 1) 119889119898minus1

119889119904119898minus1 ( 1198901205851199041199042 minus 1199012)100381610038161003816100381610038161003816100381610038161003816119904=0

(E12)

The integral around path 2 for the contour of Figure 1contributes zero due to Jordanrsquos Lemma [25] We now havethe following expression valid forall120585 isin R ge 0

119865119898 (120585) = 120585119898+1119898 + 1 minus infinsum119901=1

(minus1)119901

sdot [ 119898119890minus120585119901119901119898+1119890119898120587119894 minus 2119898 119889119898minus1119889119904119898minus1 ( 1198901205851199041199042 minus 1199012)100381610038161003816100381610038161003816100381610038161003816119904=0]

(E13)

Case 1 (119898 = 0)1198650 (120585) = 120585 minus infinsum

119901=1

(minus1)119901 119890minus120585119901119901 = ln (1 + 119890120585) (E13a)Case 2 (119898 = 1)

1198651 (120585) = 12058726 + 12058522 + infinsum119901=1

(minus1)119901 119890minus1205851199011199012 (E13b)Case 3 (119898 = 2)

1198652 (120585) = 12058721205853 + 12058533 minus 2 infinsum119901=1

(minus1)119901 119890minus1205851199011199013 (E13c)

1

x

iy2

Bromw

ich Contour

1 2minus2 minus1minus30

Figure 1 Contour used for Examples 1 and 2

The Fermi-Dirac function119865119898(120585) for any value of119898 isin Zge caneasily be evaluated Employing (5) instead of (8) for 119898 = 0yields the following

1198650 (120585)= int1205850119889119909

+ infinsum119901=1

(minus1)119901 [int1205850119890minus119901(120585minus119909)119889119909 minus intinfin

120585119890minus119901(119909minus120585)119889119909]

= 120585 + infinsum119901=1

(minus1)119901119901 (1 minus 119890minus119901120585) minus infinsum119901=1

(minus1)119901119901= ln (1 + 119890120585)

(E14)

Alternatively (5) can be employed to evaluate 119865119898(120585) for any119898 isin Zge However employing contour integration availsone of all the mathematical machineries of complex variabletheory and allows one to tackle a wide range of integralsthat may otherwise be rather complicated to evaluate byalternative methods The same analysis can be employed toanalytically evaluate the Bose-Einstein function 119861(120585) for119891(119909) = 119909119898 forall119898 isin Zge

32 Example 2 As a second example consider a slightlymorecomplicated function Let 119891(119909) = sin(119886119909) forall119886 isin R and findthe expression for the Fermi-Dirac function Employing (8)and using the contour of Figure 1 yield the following

119865 (120585) = int1205850sin (119886119909) 119889119909 + 2119886 infinsum

119901=1

(minus1)119901 12120587119894sdot int120590+119894infin120590minus119894infin

119904119890120585119904(1199042 + 1198862) (1199042 minus 1199012)119889119904


= 1 minus cos (119886120585)119886 + 2119886 infinsum119901=1

(minus1)1199011199012 + 1198862 12120587119894 [ 119904119890120585119904(1199042 minus 1199012)minus 119904119890120585119904(1199042 + 1198862)] 119889119904

= 1119886 minus 120587 cos (120585119886)sinh (120587119886) + 119886 infinsum

119901=1

(minus1)119901 119890minus1199011205851199012 + 1198862 (E21)

The same analysis can be used to compute the Bose-Einsteinfunction by employing (6) or (9)

33 Example 3 Consider an even more difficult problemSubstituting 119891(119909) = radic119909 into (1) and (2) yields the well-known half-order Fermi-Dirac function 11986512(120585) and thehalf-order Bose-Einstein function 11986112(120585) respectively (seeAppendix) The half-order Fermi-Dirac function has alreadybeen analytically evaluated by J A Selvaggi and J P Selvaggi[1] Let us attempt to analytically evaluate the half-order Bose-Einstein function forall120585 isin R ge 0 and use the result tofind an alternative expression for the half-order Fermi-Diracfunction not discussed in J A Selvaggi and J P Selvaggi [1]The half-order Bose-Einstein function is given by

11986112 (120585) = intinfin0

radic119909minus1 + 119890119909minus120585 119889119909 (E31)valid forall120585 isin R ge 0The Laplace Transform of radic119909 is radic120587211990432Employ (9) to evaluate the integral in (E31) as follows

11986112 (120585) = minus2312058532

minus radic120587 infinsum119901=1

12120587119894 int120590+119894infin


119890120585119904radic119904 (1199042 minus 1199012)119889119904(E32)

The analytical evaluation of the inversion integral givenin (E32) is expedited by employing the contour shownin Figure 2 The authors call this the Fermi-Dirac contourbecause of its utility in aiding in the analytical evaluation ofthe half-order Fermi-Dirac functions 119865minus12(120585) and 11986512(120585)Once again the Fermi-Dirac contour is closed in the left-hand plane in order to ensure convergence

Employing contour integration in the complex plane andapplying the theory of residues yield the following

12120587119894 119890120585119904radic119904 (1199042 minus 1199012)119889119904= 12120587119894 int

120590+119894infin


119890120585119904radic119904 (1199042 minus 1199012)119889119904+ 12120587119894 int

0

infin

119890119894120585119910radic1199101198901205871198942 [(1199101198901205871198942)2 minus 1199012] 119894119889119910

+ 12120587119894 intinfin

0

119890119894120585119910radic119910119890minus31205871198942 [(119910119890minus31205871198942)2 minus 1199012] 119894119889119910

(E33)

1

2

3

4

5

6

3

x

iy

Bromw

ich Contour


Figure 2 Fermi-Dirac contour

Only the line integrals along paths 1 (Bromwich Contour) 3and 5 in Figure 2 contribute a nonzero quantity to the closedline integral on the left-hand side of (E33) The closed lineintegral on the left-hand side of (E33) encloses only simplepoles since the branch point at the origin has been excludedfrom the contour of integrationThemathematical details arefound in J A Selvaggi and J P Selvaggi [1] The result is asfollows

minus 119894119890minus119901120585211990132 = 12120587119894 int120590+119894infin


119890120585119904radic119904 (1199042 minus 1199012)119889119904+ 119890minus12058711989442120587 intinfin

0

119890119894120585119910radic119910 [1199102 + 1199012]119889119910

minus 119890312058711989442120587 intinfin0

119890119894120585119910radic119910 [1199102 + 1199012]119889119910

(E34)

After simplifying the algebra the Inverse Laplace Transformbecomes

12120587119894 int120590+119894infin


119890120585119904radic119904 (1199042 minus 1199012)119889119904= minus 1211990132 [119890119901120585Erfc(radic119901120585) + 119890minus119901120585Erfi(radic119901120585)]

(E35)

Once again the mathematical details used in obtaining(E35) are given in J A Selvaggi and J P Selvaggi [1]Employing (E35) and (E32) yields the following half-orderBose-Einstein function

11986112 (120585) = minus2312058532 + radic1205872infinsum119901=1

111990132sdot [119890119901120585Erfc(radic119901120585) + 119890minus119901120585Erfi(radic119901120585)]

(E36)


The functions Erfc(∙) and Erfi(∙) are the Complimentary andImaginary Error functions respectively The evaluation of11986112(120585) forall120585 isin R lt 0 is solved by elementary methods and willnot be discussed in this article The half-order Fermi-Diracfunction is found by employing the following duplicationformula

119865120592 (120585) = 119861120592 (120585) minus 12120592119861120592 (2120585) (E37)This expression is discussed in Clunie [5] and Dingle [6]and it links the Bose-Einstein functions to the Fermi-Diracfunctions When 119891(119909) = 119909120592 and 120592 = 12 the followingexpression is found

11986512 (120585) = 11986112 (120585) minus 1radic211986112 (2120585) (E38)Employing (E38) and (E36) yields the half-order Fermi-Dirac function as follows

11986512 (120585) = 2312058532 + radic1205872infinsum119901=1

111990132 [119890119901120585Erfc(radic119901120585)minus 1radic21198902119901120585Erfc(radic2119901120585) + 119890minus119901120585Erfi(radic119901120585)minus 1radic2119890minus2119901120585Erfi(radic2119901120585)]

(E39)

Equation (E39) can easily be checked by the application of(8) In doing so one obtains [1]

11986512 (120585) = 2312058532 minus radic1205872infinsum119901=1

(minus1)11990111990132sdot [119890119901120585Erfc(radic119901120585) + 119890minus119901120585Erfi(radic119901120585)]

(E310)

Equations (E39) and (E310) yield different-lookingresults but are actually identities Either (E39) or (E310) canbe used to define the half-order Fermi-Dirac function Alter-natively (5) and (6) could be used to analytically evaluatethe half-order Fermi-Dirac function or the half-order Bose-Einstein function respectively without employing contourintegration

34 Example 4 As a final example and the most difficult ofthe four examples discussed in this article the authors willanalytically evaluate the incomplete half-order Fermi-Diracfunction [27] defined as follows

11986512 (120585 119906) = int1199060

radic1199091 + 119890119909minus120585 119889119909 (E41)valid forall120585 isin R and forall119906 isin R ge 0 Rewrite (E41) as follows

11986512 (120585 119906) = intinfin0

radic1199091 + 119890119909minus120585 119889119909 minus intinfin119906

radic1199091 + 119890119909minus120585 119889119909 (E42)= 11986512 (120585) minus 11986612 (120585 119906) (E43)

To evaluate 11986612(120585 119906) let 119910 = 119909 minus 119906 in the second integral of(E42) The result is as follows

11986612 (120585 119906) = intinfin0

radic119910 + 1199061 + 119890119910minus(120585minus119906) 119889119910 (E44)

Some care must be taken when considering (E44) There arethree cases which need to be considered These cases are asfollows

Case 1 Find 11986612(120585 119906) forall120585 isin R ge 0 forall119906 isin R le 120585Case 2 Find 11986612(120585 119906) forall120585 isin R ge 0 forall119906 isin R ge 120585Case 3 Find 11986612(120585 119906) forall120585 isin R le 0 forall119906 isin R ge 0

The authors will only analyze Case 1 since the othercases follow the same procedure One can rewrite (E42)employing (E44) as follows

11986512 (120585 119906) = 11986512 (120585) minus intinfin0


An expression for the half-order Fermi-Dirac function isgiven by (E310) Evaluate11986612(120585 119906) using (8) with F(119904) givenby

F (119904) = 2radic119904119906 + radic120587119890119904119906Erfc (radic119904119906)211990432 (E46)

Employing (E45) (E46) and (8) yields the followingexpression for 11986612(120585 119906)

11986612 (120585 119906) = 23 (12058532 minus 11990632)+ radic120587 infinsum119901=1

(minus1)119901 119868 (119906 119901) (E47)

where

119868 (119906 119901)= 12120587119894 int

120590+119894infin


2radic119904119906120587 + 119890119904119906Erfc (radic119904119906)radic119904 (1199042 minus 1199012) 119890(120585minus119906)119904119889119904 (E48)

Take note that 120585 in (8) has been replaced by 120585 minus 119906 in theevaluation of 11986612(120585 119906) This is true because the effectivedegeneracy parameter in (E45) is 120585 minus 119906 Evaluate 119868(119906 119901)by employing the Fermi-Dirac contour and substituting theresult in (E47) This yields the following

11986612 (120585 119906) = 23 (12058532 minus 11990632) + radic119906 ln [1 + 119890minus(120585minus119906)]minus radic1205872

infinsum119901=1

(minus1)11990111990132 119890120585119901Erfc(radic119901120585)minus 119890minus120585119901 [Erfi (radic119906119901) minus Erfi(radic120585119901)]

(E49)


Finally employing (E310) (E43) and (E49) yields theanalytical solution for the incomplete half-order Fermi-Diracfunction valid forall120585 isin R ge 0 and forall119906 isin R le 120585

11986512 (120585 119906) = 2311990632 minus radic119906 ln [1 + 119890minus(120585minus119906)]minus radic1205872

infinsum119901=1

(minus1)11990111990132 119890minus120585119901Erfi (radic119906119901) (E410)

The result given by (E410) is easily checked numericallyby employing Mathematica [23] Also it is in completenumerical agreement with that published by Keller andFenwick [27]The authors believe that (E410) exists nowhereelse in the published literature It is a simple matter to repeatthe process to analytically evaluate the incomplete half-orderBose-Einstein function

We leave it to the interested reader to verify that theincomplete half-order Fermi-Dirac functions for Cases 2 and3 respectively are as follows

11986512 (120585 119906) = 2312058532 minus radic119906 ln [1 + 119890minus(119906minus120585)] minus radic1205872sdot infinsum119901=1

(minus1)11990111990132 119890119901120585 [Erf (radic119906119901) minus Erf (radic120585119901)]+ 119890minus119901120585Erfi(radic119901120585)

(E411)

11986512 (120585 119906) = minusradic119906 ln [1 + 119890minus(119906+|120585|)] minus radic1205872sdot infinsum119901=1

(minus1)11990111990132 119890minus119901|120585|Erf (radic119906119901) (E412)

4 Remarks and Conclusion

This article illustrates the utility of real convolution for ana-lytically evaluating the Fermi-Dirac-type and Bose-Einstein-type functions These functions occupy a very importantrole in statistical mechanics quantum statistical mechanicssolid-state physics and others The chosen technique is ageneralization of the method developed by the authors [1]A few examples are used to illustrate the efficacy of themethod In particular the authors chose to look at the well-known half-order Fermi-Dirac and half-order Bose-Einsteinfunctions because of their importance in a number of areasof physics and engineering However the method developedin this paper is quite general and may be employed toanalytically evaluate integrals given by (1) or (2) with anarbitrary 119891(119909) Of course the functional form of 119891(119909) maybe such that the integrals in (1) or (2) will be very difficult foranalytical evaluation

It is emphasized that one may choose to directly integratethe Fermi-Dirac-type or the Bose-Einstein-type functions byemploying a direct method as exhibited by (5) or (6) or bycontour integration as exhibited by (8) or (9) The methodthat is chosen lies with the discretion of the practitionerHowever the application of real convolution and contourintegration was the main focus of the article because the

Fermi-Dirac-type functions and Bose-Einstein-type func-tions perfectly fit the form of real-convolution integrals

Although this article focused on analytical solutions afew comments should be made concerning numerical anal-ysis For example a recent article written by Mohankumarand Natarajan [28] gives a very accurate method for numeri-cally computing the Generalized Fermi-Dirac integrals Alsorecent articles by Changshi [29] and Fukushima [30] giveexcellent methods for numerically computing Fermi-Dirac-type integrals There are many other published methods thatwork very well for numerically computing the Fermi-Dirac-type and Bose-Einstein-type integrals The authors of thisarticle employed the method developed by Chevillard andRevol [31 32] and achieved promising numerical resultsHowever there will always be new and improved methodsof numerical computation and the authors simply cannotpredict with any certainty if the analytical solutions devel-oped in this paper will be of greater or lesser utility thanany other method when it comes to its use in numericalanalysis However what the authors can state with certaintyis that a very simple procedure has been developed foranalytically evaluating Fermi-Dirac-type and Bose-Einstein-type integrals for a wide range of functions 119891(119909) Thetechnique employed for doing this may lead to new ideas fornumerical formulations as well as asymptotic formulationsand may provide new computational insight

Analytical expressions can help aid in the development ofapproximate solutions [33 34] as well as in the developmentof asymptotic solutions [35] which are used extensively fornumerical computationOften a complete analytical solutioncan help to aid in the development of numerical algorithms aswell as to give the researcher insight into how the parametersof an integral fit within the form of the solution The authorsbelieve that when an analytical solution can be found evenif mathematically complex it can be of great utility In factthe analytical solutions developed in this article coupled withthe methods developed by Chevillard and Revol [31 32] haveaided the authors in the development of various algorithmsfor numerically computing the half-order Fermi-Dirac andhalf-order Bose-Einstein functions Chevillardrsquos method isbased upon a set of algorithms for numerically computingerror functions with great accuracy The results as statedearlier appear to be quite promising Further research isongoing

Onemust be cognizant however that analytical solutionsare not necessarily used for high-precision numerical compu-tations especially if the analytical solution is in the form of aninfinite series Often the series may be slower to converge forvarious values of its parameters and this requires one to finda more efficient numerical or asymptotic solution As statedearlier many such algorithms currently exist in the literature

One final point should be stressedThemethod developedin this paper has recently been employed for analyticallyevaluating the Gauss-Fermi integral [36] This integral waspreviously known to not have a complete analytical solu-tion [37] This is one more indication of the efficacy ofthe method for analytically evaluating Fermi-Dirac-type orBose-Einstein-type integrals for a variety of functions119891(119909)illustrated in (1) and (2)


Appendix

Any one of the half-order Fermi-Dirac or half-order Bose-Einstein functions enables any other half-order Fermi-Diracor half-order Bose-Einstein functions to be evaluated forall119898 isinZge This is illustrated in Dingle [3 6] as follows

F119898minus12 (120585) = 1Γ (119898 + 12) intinfin

0

119909119898minus121 + 119890119909minus120585 119889119909= 1Γ (119898 + 12)119865119898minus12 (120585)

B119898minus12 (120585) = 1Γ (119898 + 12) intinfin

0

119909119898minus12minus1 + 119890119909minus120585 119889119909= 1Γ (119898 + 12)119861119898minus12 (120585)

(A1)

Employing (A1) and the known recurrence relationships [36] given by

F119898minus12 (120585) = 119889119889120585F119898+12 (120585) B119898minus12 (120585) = 119889119889120585B119898+12 (120585)

(A2)

yields all the half-order Fermi-Dirac functions or half-orderBose-Einstein functions For example evaluate Fminus12(120585) andF32(120585) as follows

Fminus12 (120585) = 119889119889120585F12 (120585)

= 2radic 120585120587minus infinsum119901=1

(minus1)11990111990112 [119890119901120585Erfc(radic119901120585) minus 119890minus119901120585Erfi(radic119901120585)] (A3)

F32 (120585) = intF12 (120585) 119889120585

= 120587radic1205871205853 + 8120585215 radic 120585120587minus infinsum119901=1


Likewise evaluateBminus12(120585) andB32(120585) as followsBminus12 (120585) = 119889119889120585B12 (120585)

= minus2radic 120585120587+ infinsum119901=1

111990112 [119890119901120585Erfc(radic119901120585) minus 119890minus119901120585Erfi(radic119901120585)] (A5)

B32 (120585) = intB12 (120585)

= 2120587radic1205871205853 minus 8120585215 radic 120585120587+ infinsum119901=1


All half-order Fermi-Dirac and half-order Bose-Einsteinfunctions are easily evaluated using (A1) and (A2)

Disclosure

Jerry P Selvaggi is a Former Visiting Associate Professor atSUNY New Paltz New Paltz NY 12561 Jerry A Selvaggi is aRetired Professor at Gannon University Erie PA 16501

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] J A Selvaggi and J P Selvaggi ldquoThe analytical evaluation of thehalf-order Fermi-Dirac integralsrdquo Open Mathematics Journalvol 5 pp 1ndash7 2012

[2] J S Blakemore ldquoApproximations for Fermi-Dirac integralsespecially the function F12(120578) used to describe electron densityin a semiconductorrdquo Solid-State Electronics vol 25 no 11 pp1067ndash1076 1982

[3] R B Dingle ldquoThe fermi-dirac integrals I119901(120578) =(119901)minus1 intinfin

0120576119901(119890120576minus120578 + 1)minus1119889120576rdquo Applied Scientific Research vol 6

no 1 pp 225ndash239 1957[4] J McDougall and E C Stoner ldquoThe Computation of Fermi-

Dirac Functionsrdquo Philosophical Transactions of the Royal SocietyA Mathematical Physical amp Engineering Sciences vol 237 no773 pp 67ndash104 1938

[5] J Clunie ldquoOn Bose-Einstein functionsrdquo Proceedings of thePhysical Society Section A vol 67 no 7 article no 308 pp 632ndash636 1954

[6] R B Dingle ldquoThe Bose-Einstein integrals B119901(120578) =(119901)minus1 intinfin

0120576119901(119890120576minus120578minus1)minus1119889120576rdquoApplied Scientific Research SectionA

vol 6 pp 240ndash244 1957[7] K Huang Statistical Mechanics John Wiley amp Sons Inc New

York NY USA 2nd edition 1963[8] L E ReichlAModern Course in Statistical Physics Wiley-VCH

Berlin Germany 2nd edition 2004[9] R N Hill ldquoA note on certain integrals which appear in the

theory of the ideal quantum gasesrdquoAmerican Journal of Physicsvol 38 pp 1440ndash1447 1970

[10] W J Cody and H C Thacher ldquoRational Chebyshev approx-imations for fermi-dirac integrals of orders -12 12 and 32rdquoMathematics of Computation vol 21 no 97 pp 30ndash40 1967

[11] T M Garoni N E Frankel and M L Glasser ldquoCompleteasymptotic expansions of the Fermi-Dirac integrals I119901(120578) =1Γ(119901 + 1) intinfin

0120576119901(1 + 119890120576minus120578)119889120576rdquo Journal of Mathematical Physics

vol 42 no 4 pp 1860ndash1868 2001


[12] M Goano ldquoAlgorithm 745 Computation of the Completeand Incomplete Fermi-Dirac Integralrdquo ACM Transactions onMathematical Software vol 21 no 3 pp 221ndash232 1995

[13] M Goano ldquoSeries expansion of the Fermi-Dirac integral 119865119895(119909)over the entire domain of real j and xrdquo Solid-State Electronicsvol 36 no 2 pp 217ndash221 1993

[14] A J Macleod ldquoAlgorithm 779 Fermi-Dirac Functions of Order-12 12 32 52rdquoACMTransactions onMathematical Softwarevol 24 no 1 pp 1ndash12 1998

[15] NMohankumar ldquoTwo new series for the Fermi-Dirac integralrdquoComputer Physics Communications vol 176 no 11-12 pp 665ndash669 2007

[16] N Mohankumar and A Natarajan ldquoThe accurate numericalevaluation of half-order Fermi-Dirac Integralsrdquo Physica StatusSolidi (b) ndash Basic Solid State Physics vol 188 no 2 pp 635ndash6441995

[17] M A Chaudhry and A Qadir ldquoOperator representation ofFermi-Dirac and Bose-Einstein integral functions with applica-tionsrdquo International Journal of Mathematics and MathematicalSciences vol 2007 Article ID 80515 9 pages 2007

[18] EWNg C J Devine and R F Tooper ldquoChebyshev polynomialexpansion of Bose-Einstein functions of orders 1 to 10rdquoMathematics of Computation vol 23 pp 639ndash643 1969

[19] D Cvijovi ldquoNew integral representations of the polylogarithmfunctionrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Sciences vol 463 no 2080 pp 897ndash905 2007

[20] M H Lee ldquoPolylogarithms and Riemannrsquos 120577 functionrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 56no 4 pp 3909ndash3912 1997

[21] M H Lee ldquoPolylogarithmic analysis of chemical potentialand fluctuations in a 119863-dimensional free Fermi gas at lowtemperaturesrdquo Journal of Mathematical Physics vol 36 no 3pp 1217ndash1231 1995

[22] M D Ulrich W F Seng and P A Barnes ldquoSolutions tothe Fermi-Dirac Integrals in Semiconductor Physics UsingPolylogarithmsrdquo Journal of Computational Electronics vol 1 no3 pp 431ndash434 2002

[23] Wolfram Research Inc MATHEMATICA version 112 Wol-fram Research Inc Champaign Ill USA 2017

[24] I N Sneddon Fourier Transforms McGraw-Hill New YorkNY USA 1951

[25] N W McLachlan Complex Variable Theory amp TransformCalculus Cambridge University Press 2nd edition 1955

[26] B van der Pol and H Bremmer Operational Calculus Basedon the Two-Sided Laplace Integral Cambridge University Press1955

[27] G Keller andM Fenwick ldquoTabulation of the incomplete Fermi-Dirac functionsrdquoTheAstrophysical Journal vol 117 pp 437ndash4461953

[28] N Mohankumar and A Natarajan ldquoOn the very accuratenumerical evaluation of the generalized Fermi-Dirac integralsrdquoComputer Physics Communications vol 207 pp 193ndash201 2017

[29] L Changshi ldquoMore precise determination of work functionbased onFermi-Dirac distribution andFowler formulardquoPhysicaB Condensed Matter vol 444 pp 44ndash48 2014

[30] T Fukushima ldquoPrecise and fast computation of inverse Fermi-Dirac integral of order 12 by minimax rational functionapproximationrdquo Applied Mathematics and Computation vol259 pp 698ndash707 2015

[31] S Chevillard ldquoThe functions erf and erfc computed witharbitrary precision and explicit error boundsrdquo Information andComputation vol 216 pp 72ndash95 2012

[32] S Chevillard and N Revol ldquoComputation of the error functionerf in arbitrary precision with correct roundingrdquo in Proceedingsof the 8th Conference on Real Numbers and Computers pp 27ndash36 2008

[33] O N Koroleva A V Mazhukin V I Mazhukin and PV Breslavskiy ldquoAnalytical Approximation of the Fermi-DiracIntegrals of Half-Integer and Integer Ordersrdquo MathematicalModels and Computer Simulations vol 9 pp 383ndash389 2017

[34] F J FernandezVelicia ldquoHigh-precision analytic approximationsfor the Fermi-Dirac functions by means of elementary func-tionsrdquoPhysical ReviewAAtomicMolecular andOptical Physicsvol 34 no 5 pp 4387ndash4395 1986

[35] J Boersma and M L Glasser ldquoAsymptotic expansion of aclass of Fermi-Dirac integralsrdquo SIAM Journal on MathematicalAnalysis vol 22 no 3 pp 810ndash820 1991

[36] J P Selvaggi ldquoAnalytical evaluation of the charge carrier densityof organic materials with a Gaussian density of states revisitedrdquoJournal of Computational Electronics vol 17 no 1 pp 61ndash672017

[37] G Paasch and S Scheinert ldquoCharge carrier density of organicswith Gaussian density of states Analytical approximation forthe Gauss-Fermi integralrdquo Journal of Applied Physics vol 107no 10 pp 104501-1ndash104501-4 2010

Hindawiwwwhindawicom Volume 2018

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Dierential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


for these integrals for 119891(119909) = 119909120592 forall120592 gt minus1 is the Poly-logarithm function [19ndash22] This function has been studiedextensively in the literature In fact mathematical softwaresuch as Mathematica [23] uses a Polylogarithm algorithmto numerically compute the Fermi-Dirac and Bose-Einsteinintegrals However the authors will illustrate that the appli-cation of real convolution allows for the complete analyticalevaluation of the integrals in (1) and (2) for a wide rangeof functions 119891(119909) The authors have already employed thistechnique [1] to analytically evaluate the integral in (1) forthe well-known and important case of the half-order Fermi-Dirac functions where 119891(119909) = 119909119898minus12 forall119898 isin Zge A fewexamples will be considered which will help to illustratethe efficacy of the method Each solution was numericallychecked by employingMathematica [23] and other numericalalgorithms

2 Theoretical Development

Rewrite the integrals given in (1) and (2) in terms of twoconvergent real-convolution integrals [24] The observationthat real convolution can be employed is the main focus ofthis article In fact once the integrals are transformed intothe proper real-convolution form the difficulty in analyticallyevaluating (1) and (2) may be substantially reduced

In order to see that (1) and (2) can each be transformedinto two convergent real-convolution integrals requires thateach integral be put into the proper form To this end rewrite(1) and (2) as follows

119865 (120585) = int1205850

119891 (119909)1 + 119890minus(120585minus119909) 119889119909 + intinfin

120585

119891 (119909) 119890minus(119909minus120585)1 + 119890minus(119909minus120585) 119889119909 (3)

119861 (120585) = minusint1205850

119891 (119909)1 minus 119890minus(120585minus119909) 119889119909 + intinfin

120585

119891 (119909) 119890minus(119909minus120585)1 minus 119890minus(119909minus120585) 119889119909 (4)

valid forall120585 isin R ge 0The denominators in each of the integrals in (3) and (4)

can be expanded in a binomial expansion Each expansionresults in a convergent integral within its limits of integrationEmploying (3) and expanding the denominator of bothintegrals in their appropriate binomial expansion yields

119865 (120585) = int1205850119891 (119909) 119889119909 + infinsum

119901=1

(minus1)119901

sdot [int1205850119891 (119909) 119890minus119901(120585minus119909)119889119909 minus intinfin

120585119891 (119909) 119890minus119901(119909minus120585)119889119909]

(5)

Likewise (4) can be rewritten as

119861 (120585) = minusint1205850119891 (119909) 119889119909

minus infinsum119901=1

[int1205850119891 (119909) 119890minus119901(120585minus119909)119889119909 minus intinfin

120585119891 (119909) 119890minus119901(119909minus120585)119889119909]

(6)

Depending upon the complexity of the function 119891(119909)direct integration may be possible However the authors will

develop the necessary mathematical machinery employingthe Laplace Transform inversion integral and contour inte-gration of complex variable theory in order to analyticallyevaluate (5) and (6) The two integrals within the summationin (5) for example are real-convolution integrals [24] definedas follows

int1205850119891 (119909) 119890minus119901(120585minus119909)119889119909 = 12120587119894 int

120590+119894infin


F (119904) 119890120585119904(119904 + 119901) 119889119904intinfin120585

119891 (119909) 119890minus119901(119909minus120585)119889119909 = 12120587119894 int120590+119894infin


F (119904) 119890120585119904(minus119904 + 119901)119889119904(7)

where F(119904) is defined as the Laplace Transform of 119891(119909) Bothexpressions are valid forforall120585 isin R gt 0One can eventually relaxthe restriction on 120585 to include forall120585 isin R ge 0 Substituting (7)into (5) yields the following

119865 (120585) = int1205850119891 (119909) 119889119909

+ 2 infinsum119901=1

(minus1)119901 12120587119894 int120590+119894infin


119904F (119904) 119890120585119904(1199042 minus 1199012)119889119904(8)

Equation (8) represents an exact alternative expression for(1) The only restriction put upon 119891(119909) is that it must be afunction which allows the integral in (1) to be convergentOf course 119891(119909) must have a Laplace Transform The sameprocedure illustrated above yields the following expressionfor the Bose-Einstein-type integral

119861 (120585) = minusint1205850119891 (119909) 119889119909 minus 2 infinsum

119901=1

12120587119894 int120590+119894infin


119904F (119904) 119890120585119904(1199042 minus 1199012)119889119904 (9)

The first integral shown in (8) or (9) may be quite simplefor practical problems if 119891(119909) is an analytically integrablefunction The second integral shown in (8) or (9) is theinverse Laplace Transform [24ndash26] defined as follows

119892 (120585 119901) = Lminus1 [119866 (119904)] = 12120587119894 int

120590+119894infin


119904F (119904) 119890120585119904(1199042 minus 1199012)119889119904 (10a)

where

119866 (119904) = 119904F (119904)(1199042 minus 1199012) (10b)

The contour defined by the limits of integration in (10a)is called the Bromwich Contour [24ndash26] Of course there isno guarantee that the analytical evaluation of the inversionintegral of (10a) will be an easy task However for manypractical problems found in the literature this appears not tobe a problem

3 Application of Real Convolution

31 Example 1 This simple example is used to verify themethod developed in Section 2 Let 119891(119909) = 119909119898 forall119898 isin Zge



119865119898 (120585)= 120585119898+1119898 + 1

+ 2 (119898) infinsum119901=1

(minus1)119901 12120587119894 int120590+119894infin


119890120585119904119904119898 (1199042 minus 1199012)119889119904(E11)



12120587119894 119890120585119904119904119898 (1199042 minus 1199012)119889119904= 12120587119894 int

120590+119894infin


119890120585119904119904119898 (1199042 minus 1199012)119889119904= 119890120585119904119904119898 (119904 minus 119901)

100381610038161003816100381610038161003816100381610038161003816119904=119901119890120587119894+ 1(119898 minus 1) 119889

119898minus1

119889119904119898minus1 ( 1198901205851199041199042 minus 1199012)100381610038161003816100381610038161003816100381610038161003816119904=0


119889119904119898minus1 ( 1198901205851199041199042 minus 1199012)100381610038161003816100381610038161003816100381610038161003816119904=0

(E12)


119865119898 (120585) = 120585119898+1119898 + 1 minus infinsum119901=1

(minus1)119901


(E13)


119901=1


1198651 (120585) = 12058726 + 12058522 + infinsum119901=1


1198652 (120585) = 12058721205853 + 12058533 minus 2 infinsum119901=1

(minus1)119901 119890minus1205851199011199013 (E13c)

1

x

iy2

Bromw

ich Contour




1198650 (120585)= int1205850119889119909

+ infinsum119901=1


120585119890minus119901(119909minus120585)119889119909]

= 120585 + infinsum119901=1


(minus1)119901119901= ln (1 + 119890120585)

(E14)



119865 (120585) = int1205850sin (119886119909) 119889119909 + 2119886 infinsum

119901=1


119904119890120585119904(1199042 + 1198862) (1199042 minus 1199012)119889119904





119901=1

(minus1)119901 119890minus1199011205851199012 + 1198862 (E21)



11986112 (120585) = intinfin0


11986112 (120585) = minus2312058532


12120587119894 int120590+119894infin


119890120585119904radic119904 (1199042 minus 1199012)119889119904(E32)



12120587119894 119890120585119904radic119904 (1199042 minus 1199012)119889119904= 12120587119894 int

120590+119894infin


119890120585119904radic119904 (1199042 minus 1199012)119889119904+ 12120587119894 int

0

infin

119890119894120585119910radic1199101198901205871198942 [(1199101198901205871198942)2 minus 1199012] 119894119889119910

+ 12120587119894 intinfin

0


(E33)

1

2

3

4

5

6

3

x

iy

Bromw

ich Contour







0

119890119894120585119910radic119910 [1199102 + 1199012]119889119910

minus 119890312058711989442120587 intinfin0

119890119894120585119910radic119910 [1199102 + 1199012]119889119910

(E34)


12120587119894 int120590+119894infin



(E35)




(E36)





11986512 (120585) = 2312058532 + radic1205872infinsum119901=1


(E39)




(E310)



11986512 (120585 119906) = int1199060


11986512 (120585 119906) = intinfin0




11986612 (120585 119906) = intinfin0





11986512 (120585 119906) = 11986512 (120585) minus intinfin0






(minus1)119901 119868 (119906 119901) (E47)

where

119868 (119906 119901)= 12120587119894 int

120590+119894infin





infinsum119901=1


(E49)




infinsum119901=1






(E411)












Appendix



0



0


(A1)


F119898minus12 (120585) = 119889119889120585F119898+12 (120585) B119898minus12 (120585) = 119889119889120585B119898+12 (120585)

(A2)


Fminus12 (120585) = 119889119889120585F12 (120585)



F32 (120585) = intF12 (120585) 119889120585






B32 (120585) = intB12 (120585)




Disclosure




References

















vol 42 no 4 pp 1860ndash1868 2001



































Journal of












Journal of







Volume 2018






Volume 2018










119865119898 (120585)= 120585119898+1119898 + 1

+ 2 (119898) infinsum119901=1

(minus1)119901 12120587119894 int120590+119894infin


119890120585119904119904119898 (1199042 minus 1199012)119889119904(E11)



12120587119894 119890120585119904119904119898 (1199042 minus 1199012)119889119904= 12120587119894 int

120590+119894infin


119890120585119904119904119898 (1199042 minus 1199012)119889119904= 119890120585119904119904119898 (119904 minus 119901)

100381610038161003816100381610038161003816100381610038161003816119904=119901119890120587119894+ 1(119898 minus 1) 119889

119898minus1

119889119904119898minus1 ( 1198901205851199041199042 minus 1199012)100381610038161003816100381610038161003816100381610038161003816119904=0


119889119904119898minus1 ( 1198901205851199041199042 minus 1199012)100381610038161003816100381610038161003816100381610038161003816119904=0

(E12)


119865119898 (120585) = 120585119898+1119898 + 1 minus infinsum119901=1

(minus1)119901


(E13)


119901=1


1198651 (120585) = 12058726 + 12058522 + infinsum119901=1


1198652 (120585) = 12058721205853 + 12058533 minus 2 infinsum119901=1

(minus1)119901 119890minus1205851199011199013 (E13c)

1

x

iy2

Bromw

ich Contour




1198650 (120585)= int1205850119889119909

+ infinsum119901=1


120585119890minus119901(119909minus120585)119889119909]

= 120585 + infinsum119901=1


(minus1)119901119901= ln (1 + 119890120585)

(E14)



119865 (120585) = int1205850sin (119886119909) 119889119909 + 2119886 infinsum

119901=1


119904119890120585119904(1199042 + 1198862) (1199042 minus 1199012)119889119904





119901=1

(minus1)119901 119890minus1199011205851199012 + 1198862 (E21)



11986112 (120585) = intinfin0


11986112 (120585) = minus2312058532


12120587119894 int120590+119894infin


119890120585119904radic119904 (1199042 minus 1199012)119889119904(E32)



12120587119894 119890120585119904radic119904 (1199042 minus 1199012)119889119904= 12120587119894 int

120590+119894infin


119890120585119904radic119904 (1199042 minus 1199012)119889119904+ 12120587119894 int

0

infin

119890119894120585119910radic1199101198901205871198942 [(1199101198901205871198942)2 minus 1199012] 119894119889119910

+ 12120587119894 intinfin

0


(E33)

1

2

3

4

5

6

3

x

iy

Bromw

ich Contour







0

119890119894120585119910radic119910 [1199102 + 1199012]119889119910

minus 119890312058711989442120587 intinfin0

119890119894120585119910radic119910 [1199102 + 1199012]119889119910

(E34)


12120587119894 int120590+119894infin



(E35)




(E36)





11986512 (120585) = 2312058532 + radic1205872infinsum119901=1


(E39)




(E310)



11986512 (120585 119906) = int1199060


11986512 (120585 119906) = intinfin0




11986612 (120585 119906) = intinfin0





11986512 (120585 119906) = 11986512 (120585) minus intinfin0






(minus1)119901 119868 (119906 119901) (E47)

where

119868 (119906 119901)= 12120587119894 int

120590+119894infin





infinsum119901=1


(E49)




infinsum119901=1






(E411)












Appendix



0



0


(A1)


F119898minus12 (120585) = 119889119889120585F119898+12 (120585) B119898minus12 (120585) = 119889119889120585B119898+12 (120585)

(A2)


Fminus12 (120585) = 119889119889120585F12 (120585)



F32 (120585) = intF12 (120585) 119889120585






B32 (120585) = intB12 (120585)




Disclosure




References

















vol 42 no 4 pp 1860ndash1868 2001



































Journal of












Journal of







Volume 2018






Volume 2018












119901=1

(minus1)119901 119890minus1199011205851199012 + 1198862 (E21)



11986112 (120585) = intinfin0


11986112 (120585) = minus2312058532


12120587119894 int120590+119894infin


119890120585119904radic119904 (1199042 minus 1199012)119889119904(E32)



12120587119894 119890120585119904radic119904 (1199042 minus 1199012)119889119904= 12120587119894 int

120590+119894infin


119890120585119904radic119904 (1199042 minus 1199012)119889119904+ 12120587119894 int

0

infin

119890119894120585119910radic1199101198901205871198942 [(1199101198901205871198942)2 minus 1199012] 119894119889119910

+ 12120587119894 intinfin

0


(E33)

1

2

3

4

5

6

3

x

iy

Bromw

ich Contour







0

119890119894120585119910radic119910 [1199102 + 1199012]119889119910

minus 119890312058711989442120587 intinfin0

119890119894120585119910radic119910 [1199102 + 1199012]119889119910

(E34)


12120587119894 int120590+119894infin



(E35)




(E36)





11986512 (120585) = 2312058532 + radic1205872infinsum119901=1


(E39)




(E310)



11986512 (120585 119906) = int1199060


11986512 (120585 119906) = intinfin0




11986612 (120585 119906) = intinfin0





11986512 (120585 119906) = 11986512 (120585) minus intinfin0






(minus1)119901 119868 (119906 119901) (E47)

where

119868 (119906 119901)= 12120587119894 int

120590+119894infin





infinsum119901=1


(E49)




infinsum119901=1






(E411)












Appendix



0



0


(A1)


F119898minus12 (120585) = 119889119889120585F119898+12 (120585) B119898minus12 (120585) = 119889119889120585B119898+12 (120585)

(A2)


Fminus12 (120585) = 119889119889120585F12 (120585)



F32 (120585) = intF12 (120585) 119889120585






B32 (120585) = intB12 (120585)




Disclosure




References

















vol 42 no 4 pp 1860ndash1868 2001



































Journal of












Journal of







Volume 2018






Volume 2018












11986512 (120585) = 2312058532 + radic1205872infinsum119901=1


(E39)




(E310)



11986512 (120585 119906) = int1199060


11986512 (120585 119906) = intinfin0




11986612 (120585 119906) = intinfin0





11986512 (120585 119906) = 11986512 (120585) minus intinfin0






(minus1)119901 119868 (119906 119901) (E47)

where

119868 (119906 119901)= 12120587119894 int

120590+119894infin





infinsum119901=1


(E49)




infinsum119901=1






(E411)












Appendix



0



0


(A1)


F119898minus12 (120585) = 119889119889120585F119898+12 (120585) B119898minus12 (120585) = 119889119889120585B119898+12 (120585)

(A2)


Fminus12 (120585) = 119889119889120585F12 (120585)



F32 (120585) = intF12 (120585) 119889120585






B32 (120585) = intB12 (120585)




Disclosure




References

















vol 42 no 4 pp 1860ndash1868 2001



































Journal of












Journal of







Volume 2018






Volume 2018











infinsum119901=1






(E411)












Appendix



0



0


(A1)


F119898minus12 (120585) = 119889119889120585F119898+12 (120585) B119898minus12 (120585) = 119889119889120585B119898+12 (120585)

(A2)


Fminus12 (120585) = 119889119889120585F12 (120585)



F32 (120585) = intF12 (120585) 119889120585






B32 (120585) = intB12 (120585)




Disclosure




References

















vol 42 no 4 pp 1860ndash1868 2001



































Journal of












Journal of







Volume 2018






Volume 2018









Appendix



0



0


(A1)


F119898minus12 (120585) = 119889119889120585F119898+12 (120585) B119898minus12 (120585) = 119889119889120585B119898+12 (120585)

(A2)


Fminus12 (120585) = 119889119889120585F12 (120585)



F32 (120585) = intF12 (120585) 119889120585






B32 (120585) = intB12 (120585)




Disclosure




References

















vol 42 no 4 pp 1860ndash1868 2001



































Journal of












Journal of







Volume 2018






Volume 2018










































Journal of












Journal of







Volume 2018






Volume 2018








the application of real convolution for analytically...

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