unusual Fourier Series
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A reference compendium by John Washburn, dated April 30, 2010, apparently kept in Phil's archive as a collected source. It lists closed-form functions f(t) for Fourier coefficients such as 1/(n+α), 1/(n(n+α)) and n/(n²±α²), grouped by numerator and denominator degree. Later sections are mostly marked to-do, and an appendix gives partial fraction expansions. The text is somewhat garbled in places.
AI-written summary; may contain errors. This description is approximate.
Extracted text (machine-read; may contain errors)
TABLE OF UNUSUAL FOURIER SERIES
JOHN WASHBURN
Abstract. A simple compendium of Fourier series I have run
across in my research. Most functions are R!C. This is because
the function, f(t), was chosen to create a particular set of Fourier
coecients; Fourier coecients that are rational polynomials in n
of the form:Q(n)
P(n).
The organization of the functions is by the Fourier coecients;
most of which are rational polynomials in n. The organization of
the rational polynomials in nfollows the organization of polynomi-
als found in A Table of Series and Products by Eldon R. Hansen.
Date : April 30, 2010.
Key words and phrases. Fourier Series.
1
2 JOHN WASHBURN
Contents
1. Introduction 3
2. r=1,s=0 5
3. r=2, s=0 6
4. r=2,s=1 8
5. r=3,s=0 9
6. r=3,s=1 10
7. r=3,s=2 11
Appendix A. Table of Partial Fractions 12
A.1. r=2 12
A.2. r=3 13
References 15
TABLE OF UNUSUAL FOURIER SERIES 3
1.Introduction
The organization of table follows the organization of A Table of Series and Products [1]
by Eldon R. Hansen. For a periodic function, f(t), with a period of,
P, the Fourier series representation will generally be of the form:
f(t) =X
n2Zcne2int
P
=X
n2ZQ(n)
P(n)e2int
P
WhereQ(n) is a polynomial of degree sandP(n) is a polynomial
of degreer. The organization of the Fourier coecients, cn, which
are rational polynomials will follow the organizational example found
in chapter two, Series Involving Rational, Factorial, and Power Func-
tions.
A bene t of this organizational approach is the A Table of Series and Products [1]
by Eldon R. Hansen can be used to double check the functions, f(t)
found in this table. By using the fact that:
f(0+) +f(0 )
2=f(0+) +f(P )
2=X
n2ZQ(n)
P(n)
one can lookup the corresponding series in the Hansen Table and verify
the function, f(t), cited in this table agrees with the more authoritative
value found in the Hansen table.
For example, take the Fourier series over the range 0 <t<P :
X
n2Z1
(n+ )e2int
P=e2i (1
2 t
P)
sin ( )
Taken at the limits, 0+andP yields:
X
n2Z1
(n+ )=
sin ( )ei +e i
2
=cos ( )
sin ( )
=cot ( )
which agrees with the Hansen table entry: ????.
4 JOHN WASHBURN
Unless otherwise noted, the parameters of the functions cited in this
table are subject to the following conditions.
P;t;x; ; ;
2R
; ;
=2Z
; ;
6= 0
; ;
are distinct
P > 0
Unless otherwise noted, the summation is over all integers or over
all integers except zero. These summations over a variable nwill be
noted are follows:X
n2Z=X
n
X
n2Z
n6=0=0X
n
For a real variable, x,bxcis the
oor of xandfxgis the fractional
part ofx. Thus, for all real x,
x=bxc+fxgwhere: 0fxg<1
Functions with complicated rational, Fourier coecients were con-
structed from functions with simpler rational, Fourier coecients using
partial fractions. For example the function with Fourier coecients of
1
n(n+ )was constructed by combining the function with Fourier coef-
cients of1
nwith the the function with Fourier coecients of1
(n+ ).
Because of the extensive use of partial fractions, an appendix of partial
fraction expansions is included.
TABLE OF UNUSUAL FOURIER SERIES 5
2.r=1,s=0
X
n1
(n+ )e2int
P=e2i (1
2 t
P)
sin ( )
where: 0<t<P(1)
X
n( 1)n
(n+ )e2int
P=
sin ( )e 2i t
P
where: P
2<t<P
2(2)
0X
n1
ne2int
P= (2i)1
2 t
P
where: 0<t<P(3)
0X
n( 1)n
ne2int
P= 2it
P
where: P
2<t<P
2(4)
6 JOHN WASHBURN
3.r=2, s=0
0X
n1
n2e2int
P= 22"t
P2
t
P+1
6#
where: 0<t<P(5)
0X
n( 1)n
n2e2int
P= 22"t
P2
t
12#
where: P
2<t<P
2(6)
0X
n1
n(n+ )e2int
P="
1
2+2i
1
2 t
P
e2i (1
2 t
P)
sin ( )#
where: 0<t<P(7)
0X
n( 1)n
n(n+ )e2int
P="
1
2+ (2i)t
P e 2i t
P
sin ( )#
where: P
2<t<P
2(8)
X
ne2int
P
(n+ ) (n+ )=
( )"
e2i (1
2 t
P)
sin ( ) e2i (1
2 t
P)
sin ( )#
where: 0<t<P(9)
X
n( 1)ne2int
P
(n+ ) (n+ )=
( )"
e 2i t
P
sin ( ) e 2i t
P
sin ( )#
where: P
2<t<P
2(10)
X
n1
(n+ )2e2int
P=
1 (2i)tsin ( )
Pe i 2e2i t
P
sin2( )
where: 0<t<P(11)
X
n1
(n )2e2int
P=
1 +(2i)tsin ( )
Pei 2e 2i t
P
sin2( )
where: 0<t<P(12)
TABLE OF UNUSUAL FOURIER SERIES 7
X
n( 1)n
(n+ )2e2int
P=(2i)tsin ( )
P+ cos ( )2e 2i t
P
sin2( )
where: P
2<t<P
2(13)
X
nne2int
P
(n+ ) (n+ )=
( )"
e2i (1
2 t
P)
sin ( ) e2i (1
2 t
P)
sin ( )#
where: 0<t<P(14)
X
n( 1)nne2int
P
(n+ ) (n+ )=
( )"
e 2i t
P
sin ( ) e 2i t
P
sin ( )#
where: P
2<t<P
2(15)
XX
n1
(n2 2)e2int
P= cos
2 1
2 t
P
sin ( )
where: 0<t<P(16)
XX
n( 1)n
(n2 2)e2int
P= cos
2 t
P
sin ( )
where: P
2<t<P
2(17)
XX
n1
(n2+ 2)e2int
P=cosh
2 1
2 t
P
sinh ( )
where: 0<t<P(18)
XX
n( 1)n
(n2+ 2)e2int
P=cosh
2 t
P
sinh ( )
where: P
2<t<P
2(19)
8 JOHN WASHBURN
4.r=2,s=1
X
nne2int
P
(n+ ) (n+ )=
( )"
e2i (1
2 t
P)
sin ( ) e2i (1
2 t
P)
sin ( )#
where: 0<t<P(20)
X
nn( 1)ne2int
P
(n+ ) (n+ )=
( )"
e 2i t
P
sin ( ) e 2i t
P
sin ( )#
where: P
2<t<P
2(21)
???X
nn
(n2 2)e2int
P=isin
2 1
2 t
P
2 sin ( )
where: P
2<t<P
2(22)
???X
n( 1)nn
(n2 2)e2int
P=isin
2 t
P
2 sin ( )
where: 0<t<P(23)
???X
nn
(n2+ 2)e2int
P=isinh
2 1
2 t
P
2 sinh ( )
where: P
2<t<P
2(24)
???X
n( 1)nn
(n2+ 2)e2int
P=isinh
2 t
P
2 sinh ( )
where: 0<t<P(25)
To do:
n
(n+ )2
TABLE OF UNUSUAL FOURIER SERIES 9
5.r=3,s=0
To do:1
n3
1
n2(n+ )
1
n(n+ )2
1
(n+ )3
1
(n+ ) (n+ )2
1
n(n+ ) (n+ )
1
(n+ ) (n+ ) (n+
)
1
n(n2 2)
1
n(n2+ 2)
1
(n+ ) (n2 2)
1
(n+ ) (n2+ 2)
1
(n3+ 3)
10 JOHN WASHBURN
6.r=3,s=1
To do:
n
(n+ )3
n
(n+ ) (n+ )2
n
(n+ ) (n+ ) (n+
)
n
n(n2 2)
n
n(n2+ 2)
n
(n+ ) (n2 2)
n
(n+ ) (n2+ 2)
n
(n3+ 3)
TABLE OF UNUSUAL FOURIER SERIES 11
7.r=3,s=2
To do:
n2
(n+ )3
n2
(n+ ) (n+ )2
n2
(n+ ) (n+ ) (n+
)
n2
n(n2 2)
n2
n(n2+ 2)
n2
(n+ ) (n2 2)
n2
(n+ ) (n2+ 2)
n2
(n3+ 3)
12 JOHN WASHBURN
Appendix A.Table of Partial Fractions
A.1.r=2.
X1
n(n+ )=1
1
n 1
(n+ )
X1
(n+ ) (n+ )=1
( )1
(n+ ) 1
(n+ )
Xn
(n+ ) (n+ )=1
( )
(n+ )
(n+ )
X1
(n2 2)=1
2 1
n 1
n+
Xn
(n2 2)=1
21
n +1
n+
X1
(n2+ 2)=1
2i 1
n i 1
n+i
Xn
(n2+ 2)=1
21
n i +1
n+i
TABLE OF UNUSUAL FOURIER SERIES 13
A.2.r=3.
X1
n(n+ ) (n+ )=1
( )( )
n+
(n+ )
(n+ )
X1
n2(n+ )=1
2
n2 1
n+1
(n+ )
X1
n(n+ )2=1
21
n 1
(n+ )
(n+ )2
X1
n(n2 2)= 1
2 22
n 1
(n ) 1
(n+ )
X1
n(n2+ 2)=1
2 22
n 1
(n i ) 1
(n+i )
Xnk
(n+ ) (n+ ) (n+
)=1
( ) (
) (
)2
666666664+( )k(
)
(n+ )
( )k(
)
(n+ )
+(
)k( )
(n+
)3
777777775
where: 0k2
X1
(n+ ) (n+ )2=1
( )21
(n+ ) 1
(n+ )+( )
(n+ )2
Xn
(n+ ) (n+ )2= 1
( )2
(n+ )
(n+ )+ ( )
(n+ )2
???n2
(n+ ) (n+ )2=1
( )2 2
(n+ ) (2 )
(n+ )+ 2( )
(n+ )2
14 JOHN WASHBURN
???1
( 2 2))???1
(n+ ) (n2 2)=1
2 ( 2 2)( + )
(n+ ) ( )
(n ) 2
(n+ )
???1
( 2 2))???n
(n+ ) (n2 2)= 1
2 ( 2 2)( + )
(n+ )+( )
(n ) 2
(n+ )
???1
( 2 2))???n2
(n+ ) (n2 2)=1
2 ( 2 2)( + )
(n+ ) ( )
(n ) 2 2
(n+ )
???1
(n+ ) (n2+ 2)=1
2i ( 2+ 2)( +i )
(n+ ) ( i )
(n ) 2i
(n+ )
???n
(n+ ) (n2+ 2)= 1
2 ( 2+ 2)( +i )
(n+ )+( i )
(n ) 2
(n+ )
???n2
(n+ ) (n2+ 2)=1
2i( 2+ 2)( +i )
(n+ ) ( i )
(n )+2i 2
(n+ )
TABLE OF UNUSUAL FOURIER SERIES 15
References
[1] Eldon R. Hansen, "A Table of Series and Products"
Prentice Hall (May 1975), ISBN-10: 0138819386, ISBN-13: 978-0138819385
[2] This Paper. [math.NT]
N128W12795 Highland Road, Germantown, WI 53022
URL :http://www.WashburnResearch.org
E-mail address :[email protected]