General Addition Formula for Meromorphic Functions Derived from Residue Theorem
A meromorphic function is characterized by its singularity, and residues of the poles give us various information regarding the function. In the paper the algebraic relations of residues on some meromorphic functions are considered and a generic form of addition formula is induced from them. As the...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2010
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irk-123456789-1066392016-10-02T03:02:52Z General Addition Formula for Meromorphic Functions Derived from Residue Theorem Yosh, H. A meromorphic function is characterized by its singularity, and residues of the poles give us various information regarding the function. In the paper the algebraic relations of residues on some meromorphic functions are considered and a generic form of addition formula is induced from them. As the applications of that formula, the addition formula for rational function, hypergeometric function, elliptic function, and Riemann's Zeta function are studied. Also the general multiplication formula is discussed based on the previous discourse. 2010 Article General Addition Formula for Meromorphic Functions Derived from Residue Theorem / H. Yosh // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 2. — С. 183-191. — Бібліогр.: 7 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106639 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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A meromorphic function is characterized by its singularity, and residues of the poles give us various information regarding the function. In the paper the algebraic relations of residues on some meromorphic functions are considered and a generic form of addition formula is induced from them. As the applications of that formula, the addition formula for rational function, hypergeometric function, elliptic function, and Riemann's Zeta function are studied. Also the general multiplication formula is discussed based on the previous discourse. |
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Article |
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Yosh, H. |
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Yosh, H. General Addition Formula for Meromorphic Functions Derived from Residue Theorem Журнал математической физики, анализа, геометрии |
| author_facet |
Yosh, H. |
| author_sort |
Yosh, H. |
| title |
General Addition Formula for Meromorphic Functions Derived from Residue Theorem |
| title_short |
General Addition Formula for Meromorphic Functions Derived from Residue Theorem |
| title_full |
General Addition Formula for Meromorphic Functions Derived from Residue Theorem |
| title_fullStr |
General Addition Formula for Meromorphic Functions Derived from Residue Theorem |
| title_full_unstemmed |
General Addition Formula for Meromorphic Functions Derived from Residue Theorem |
| title_sort |
general addition formula for meromorphic functions derived from residue theorem |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/106639 |
| citation_txt |
General Addition Formula for Meromorphic Functions Derived from Residue Theorem / H. Yosh // Журнал математической физики, анализа, геометрии. — 2010. — Т. 6, № 2. — С. 183-191. — Бібліогр.: 7 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT yoshh generaladditionformulaformeromorphicfunctionsderivedfromresiduetheorem |
| first_indexed |
2025-07-07T18:48:15Z |
| last_indexed |
2025-07-07T18:48:15Z |
| _version_ |
1837015075010904064 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2010, vol. 6, No. 2, pp. 183–191
General Addition Formula for Meromorphic Functions
Derived from Residue Theorem
Harry Yosh
Department of Information Science
University of Canberra, ACT 2601, Canberra, Australia
E-mail:square17320508@yahoo.com
Received September 9, 2009
A meromorphic function is characterized by its singularity, and residues
of the poles give us various information regarding the function. In the pa-
per the algebraic relations of residues on some meromorphic functions are
considered and a generic form of addition formula is induced from them. As
the applications of that formula, the addition formula for rational function,
hypergeometric function, elliptic function, and Riemann’s Zeta function are
studied. Also the general multiplication formula is discussed based on the
previous discourse.
Key words: addition formula, multiplication formula, hypergeometric
function, elliptic function, Riemann’s Zeta function.
Mathematics Subject Classification 2000: 30D30.
Introduction
When f(z) is holomorphic on the simply connected domain D surrounded
with rectifiable curve, and z = a, b, and c are arbitrarily selected points in that
domain, the following equation is satisfied according to Cauchy’s residue theorem:
∮
∂D
f(z)
(z − a)(z − b)(z − c)
dz = 2πi
∑
x=a,b,c
Res [
f(z)
(z − a)(z − b)(z − c)
, x ]
= 2πi
(
f(a)
(a− b)(a− c)
+
f(b)
(b− a)(b− c)
+
f(c)
(c− a)(c− b)
)
. (1)
c© Harry Yosh, 2010
Harry Yosh
Choosing c as c = a + b, the above equation is rewritten as
∮
∂D
f(z)
(z − a)(z − b)(z − a− b)
dz
= 2πi
(
f(a)
(a− b)(−b)
+
f(b)
(b− a)(−a)
+
f(a + b)
ab
)
. (2)
Hence,
f(a + b) =
af(a)
a− b
+
bf(b)
b− a
+
ab
2πi
∮
∂D
f(z)
(z − a)(z − b)(z − a− b)
.dz. (3)
Since a and b are chosen arbitrarily, the above equation is satisfied all over the
domain D. Namely it gives the algebraic relation among f(a), f(b), f(a + b), a,
b, and the contour integral for any a and b (a 6= b) in D.
Now we suppose the case that the integral
∮
∂D
f(z)
(z − a)(z − b)(z − a− b)
dz (4)
converges to zero when the domain D is extended to the entire complex plane,
and f(z) is meromorphic and has the poles p1, p2, . . . , pn in the entire complex
plane (where n might be zero or infinity.) In this case the integral (4) is expressed
as
∮
∂D
f(z)
(z − a)(z − b)(z − a− b)
dz
= 2πi
(
f(a)
(a− b)(−b)
+
f(b)
(b− a)(−a)
+
f(a + b)
ab
)
+2πi
∑
j=1,2,...,n
Res [
f(z)
(z − a)(z − b)(z − c)
, pj ] → 0. (5)
After taking the limitation, f(a + b) is expressed as
f(a + b) =
af(a)− bf(b)
a− b
− ab
n∑
j=1
Res [
f(z)
(z − a)(z − b)(z − a− b)
, pj ]. (6)
Here the equation (6) is called a generic form of addition formula, although strictly
saying, it is not addition formula in the traditional meaning since it includes
not only f(a), f(b), f(a + b), but also function’s arguments a, b. This formula
suggests that the algebraic relation of f(a+ b), f(a), f(b), a, and b is determined
184 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2
General Addition Formula for Meromorphic Functions
solely by the poles of f(z). For the meromorphic functions which do not satisfy
the condition expressed with the integral (4) it is necessary to transform those
functions appropriately so as to apply the generic form of addition formula. In the
following sections, we discuss the feasibility of that formula on rational function,
hypergeometric function, elliptic function, and Riemann’s Zeta function.
Addition Formula for Rational Function
Although the addition theorem of rational functions is not new, here we see
how the generic form of addition formula (6) works on some rational functions in
the different manner, i.e., including function’s arguments.
When f(z) is a rational function, the integral (4) converges to zero if the order
of f(z) is less than two. Therefore the generic form of addition formula (6) can
be applicable to the rational functions which have the order less than two. For
example, applying the formula (6) to the function f(z) = z, we obtain
f(a + b) =
af(a)− bf(b)
a− b
(7)
since f(z) = z does not have any poles. As another example, the addition formula
for f(z) = 1
z is derived from (6) as
f(a + b) =
af(a)− bf(b)
a− b
+
1
a + b
(8)
since f(z) = 1
z has a pole at z = 0 and its residue is 1. Those formulae are easily
verified by substituting z or 1
z for f(z).
Functional Equation for Hypergeometric Function
The addition theorems of some hypergeometric type functions such as Bessel
function are obtained by applying Fourier-cosine expansion explicitly [2] although
in some cases their expressions are analytic rather than algebraic.
Here we investigate the feasibility of the generic form of addition formula (6) to
Gauss’ Hypergeometric function
2F1(a, b; c; z) = 1 +
ab
1 · c z +
a(a + 1)b(b + 1)
1 · 2 · c(c + 1)
z2 + . . . . (9)
The above expression converges in the unit disk [3]. Namely 2F1(a, b; c; 1
z ) is de-
termined uniquely on the entire complex plane except on the unit disk. Although
2F1(a, b; c; 1
z ) is multi-valued in the unit disk when 1− c, c− a− b or a− b is not
integer owing to analytic continuation, the integral (4) converges to zero when
the domain D is extended to the entire complex plane as 2F1(a, b; c; 1
z ) converges
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 185
Harry Yosh
to 1 when z →∞. Therefore we can apply the generic form of addition formula
(6) to 2F1(a, b; c; 1
z ).
As mentioned above, 2F1(a, b; c; 1
z ) is multi-valued on the unit disk when 1 − c,
c− a− b or a− b is not integer and the series
2F1(a, b; c;
1
z
) = 1 +
ab
1 · c
1
z
+
a(a + 1)b(b + 1)
1 · 2 · c(c + 1)
1
z2
+ ... (10)
does not converge there. However the integral
∮
∂D
2F1(a, b; c;
1
z
) dz
is easily calculated as long as the domain D includes the unit disk since the
integral for every term in the series (10) is zero except the second term: ab
1·c
1
z
owing to the residue theorem, i.e.
∮
∂D
2F1(a, b; c;
1
z
) dz = 2πi
ab
c
. (11)
Thus we consider z = 0 as the pole of 2F1(a, b; c; 1
z ) and its residue is ab
c . Applying
the generic form of addition formula (6) to 2F1(a, b; c; 1
z ), we obtain the following
additive functional equation for 2F1(a, b; c; 1
z ):
2F1(a, b; c;
1
x + y
)
=
x · 2F1(a, b; c; 1
x)− y · 2F1(a, b; c; 1
y )
x− y
− xy Res [ 2F1(a, b; c; 1
z )
(z − x)(z − y)(z − x− y)
, z = 0 ]
=
x · 2F1(a, b; c; 1
x)− y · 2F1(a, b; c; 1
y )
x− y
+
ab
c(x + y)
. (12)
This equation is valid for any x, y (x 6= y), and x + y outside of the unit disk.
Note that it is not the addition formula for hypergeometric function itself, but
for the hypergeometric function substituted 1
z for z.
Addition Formula for Elliptic Function
There have been known various kinds of addition theorem of elliptic functions
since that of lemniscate function by Gauss [6]. Here we discuss the addition theo-
rem which includes function’s arguments as well as the case of rational functions
186 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2
General Addition Formula for Meromorphic Functions
mentioned in the previous section. Since the elliptic function is meromorphic and
the integral (4) converges to zero when the domain D is extended to the entire
complex plane, the application of generic form of addition formula (6) to elliptic
function is straightforward.
When an elliptic function f(z) has the fundamental periods ω1, ω2, and poles p1,
p2, . . . , pd on its fundamental parallelogram, any pole of f(z) is expressed as
z = pj + nω1 + mω2 (n,m : any integers, j : 1, 2, . . . , d). (13)
Therefore the following addition formula for the elliptic function is derived from
(6):
f(x + y) =
xf(x)− yf(y)
x− y
−xy
∞∑
n,m=−∞
d∑
j=1
Res [
f(z)
(z − x)(z − y)(z − x− y)
, pj + nω1 + mω2 ]. (14)
This formula is valid for any x, y (x 6= y) on the entire complex plane except on
the poles.
Addition Formula and Functional Equation for Riemann’s Zeta
Function
Regarding the duplication formula for Riemann’s Zeta function, it is known
that ζ(2s)/ζ(s) for arbitrary s is expressed as a Dirichlet series for Liouville
function [7]. However we can not predict the algebraic relation of ζ(2s) and ζ(s)
using that formula at this stage because of the nature of Liouville function. Here
we discuss the feasibility of the generic form of addition formula (6) to Riemann’s
Zeta function for expressing algebraic additive functional relations specific to the
function.
Riemann’s Zeta function ζ(s) is a meromorphic function, therefore 1
ζ(s) is also
meromorphic. From Riemann’s functional equation [4]
ζ(s) = 2sπs−1 sin
(πs
2
)
Γ(1− s)ζ(1− s) (15)
we know 1
ζ(s) converges to zero rapidly when Re(s) ≤ 1 and |Im(s)| → ∞ except
s is integer. As ζ(s) converges to a finite value when Re(s) > 1, the integral (4)
for f(z) = 1
ζ(z) converges to zero when the domain D is extended to the entire
complex plane, and we can apply the generic form of addition formula (6) to 1
ζ(s) .
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 187
Harry Yosh
Since ζ(s) has trivial zeros and nontrivial zeros, 1
ζ(s) has trivial poles and non-
trivial poles accordingly. Regarding the trivial poles z = −2,−4,−6, . . ., their
residues are easily calculated. As each trivial pole is the first order, the residue
is expressed as
Res [
1
ζ(s)
,−2n] = lim
s→−2n
s + 2n
ζ(s)
=
1
ζ ′(s)
|s=−2n. (16)
Using Riemann’s functional equation (15),
1
ζ ′(s)
|s=−2n
=
π
π
2 cos
(
πs
2
)
(2π)sΓ(1− s)ζ(1− s) + sin
(
πs
2
)
(...)
|s=−2n
=
(−1)nπ
π
2 (2π)−2nΓ(2n + 1)ζ(2n + 1)
. (17)
Therefore when the set of nontrivial poles is denoted as P , the addition formula
for 1
ζ(s) is derived from the generic form of addition formula (6) as
1
ζ(x + y)
=
x
ζ(x) − y
ζ(y)
x− y
+
∞∑
n=1
(−1)n2(2π)2nxy
(2n + x)(2n + y)(2n + x + y)Γ(2n + 1)ζ(2n + 1)
− xy
∑
p∈P
Res [
f(z)
(z − x)(z − y)(z − x− y)
, p ]. (18)
It is required for calculating ζ(x+y) from given values of ζ(x) and ζ(y) using the
above addition formula to know the distribution of nontrivial poles (or nontrivial
zeros for Zeta function) and their residues previously. Using the transformation
similar to that applied to hypergeometric function in the previous section, we can
induce the functional equation for Zeta function without referring to nontrivial
zeros as the following.
Firstly we expand Zeta function as Laurent series about z = 1 as [5]
ζ(s) =
1
s− 1
+ γ0 − γ1
1!
(s− 1) +
γ2
2!
(s− 1)2 − γ3
3!
(s− 1)3 + · · · , (19)
where γj is Stieltjes constants defined as
γj = lim
m→∞ (
m∑
k=1
(log k)j
k
− (log m)j+1
j + 1
). (20)
188 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2
General Addition Formula for Meromorphic Functions
Transforming s to 1
s + 1, we obtain the following expression:
ζ(
1
s
+ 1) = s + γ0 − γ1
1!
1
s
+
γ2
2!
1
s2
− γ3
3!
1
s3
+ · · · . (21)
It follows that the pole of ζ(1
s + 1) is s = 0 and its residue is −γ1. When putting
f(z) = ζ(1
z + 1), the integral (4) converges to zero as the highest order of the
above series is one. Therefore we can apply the generic form of addition formula
(6) to ζ(1
z + 1) as
ζ(
1
x + y
+ 1) =
xζ( 1
x + 1)− yζ( 1
y + 1)
x− y
− γ1
x + y
. (22)
This functional equation is satisfied by any x, y (x 6= y) on the entire complex
plane except the origin.
General Multiplication Formula for Meromorphic Functions
By putting c = ab in the equation (1) instead of c = a + b the generic form
of multiplication formula for meromorphic functions is induced with the manner
similar to that discussed in Introduction. In this case, the equation (1) is rewritten
as
∮
∂D
f(z)
(z − a)(z − b)(z − ab)
dz
= 2πi
(
f(a)
(a− b)(1− b)a
+
f(b)
(b− a)(1− a)b
+
f(ab)
(b− 1)(a− 1)ab
)
, (23)
where f(z) is holomorphic on the domain D and a, b are arbitrarily selected points
in D. Therefore when f(z) is meromorphic on the entire complex plane and its
poles are p1, p2, . . . , pn, the following generic form of multiplication formula:
f(ab) =
(a− 1)bf(a)− (b− 1)af(b)
a− b
−(a− 1)(b− 1)ab
∑
j=1,2,...,n
Res [
f(z)
(z − a)(z − b)(z − ab)
, pj ] (24)
is deduced under the condition that the integral
∮
∂D
f(z)
(z − a)(z − b)(z − ab)
dz (25)
converges to zero when D is extended to the entire complex plane. Since this
condition is actually equivalent with that of the integral (4), multiplicative formu-
lae or functional equations for rational function, hypergeometric function, elliptic
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 189
Harry Yosh
function, and Zeta function are deduced from the discussion parallel with that
for deducing addition formulae for those functions.
In the case of hypergeometric function, the following multiplicative functional
equation is deduced from the generic form of multiplication formula (24):
2F1(a, b; c;
1
xy
)
=
(x− 1)y · 2F1(a, b; c; 1
x)− (y − 1)x · 2F1(a, b; c; 1
y )
x− y
+
ab(x− 1)(y − 1)
cxy
. (26)
Similarly, the multiplication formula for elliptic function is expressed as
f(xy) =
(x− 1)yf(x)− (y − 1)xf(y)
x− y
− (x− 1)(y − 1)xy
×
∞∑
n,m=−∞
d∑
j=1
Res [
f(z)
(z − x)(z − y)(z − xy)
, pj + nω1 + mω2 ], (27)
and the multiplicative functional equation for Zeta function is
ζ(
1
xy
+ 1) =
(x− 1)yζ( 1
x + 1)− (y − 1)xζ( 1
y + 1)
x− y
− γ1(x− 1)(y − 1)
xy
, (28)
where the definitions of ω1, ω2, pj , and γ1 are the same as those defined in the
previous sections.
Summary
The generic form of addition and multiplication formula for the meromorphic
functions satisfying a certain condition were induced from the residue theorem.
Those formulae include function’s arguments, and are determined solely by func-
tion’s singularities.
For the functions which do not satisfy the said condition it is often possible to
induce the additive or multiplicative functional equation instead of addition or
multiplication formula by applying the generic form of addition or multiplication
formula to the functions modified in order to satisfy the said condition.
As practical cases, the addition formula for some rational functions, elliptic func-
tion, Riemann’s Zeta function, and the additive functional equation for hyperge-
ometric function, Zeta function were deduced. Also the multiplication formula or
multiplicative functional equation for elliptic function, hypergeometric function,
and Zeta function were shown.
190 Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2
General Addition Formula for Meromorphic Functions
References
[1] E. Picard, Leçons sur quelques équations fonctionnelles. Gauthier-Villars, Paris,
1928.
[2] G.N. Watson, Theory of Bessel functions. (2nd Ed.) Cambridge Univ. Press, Cam-
bridge, 1944.
[3] A. Segun and M. Abramowitz, Handbook of Mathematical Functions. — Nat. Bur.
Standards, Appl. Math. Ser. 55 (1970).
[4] E.C. Titchmarsh, The Theory of the Riemann Zeta Function. (2nd Ed.) Oxford
University Press, Oxford, 1986.
[5] É.P. Stankus, A Note Concerning the Coefficients of the Laurent Series of the
Riemann Zeta Function. — J. Math. Sci. Springer, New York 29 (1985), No. 3,
1302–1305.
[6] N.I. Akhiezer, Elements of the theory of elliptic functions. Amer. Math. Soc.,
Providence, RI, 1990.
[7] R.S. Lehman, On Liouville’s Function. — Math. Comput. 14 (1960), 311–320.
Journal of Mathematical Physics, Analysis, Geometry, 2010, vol. 6, No. 2 191
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