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Research and Communications in Mathematics and Mathematical Sciences
Vol. 12, Issue 1, 2020, Pages 1-15
ISSN 2319-6939
Published Online on January 28, 2020
2020 Jyoti Academic Press
http://jyotiacademicpress.org
2020 Mathematics Subject Classification: 3002, 30A99, 30E15.
Keywords and phrases: polynomial, rational, inverse function, Lambert root analytic
function, Riemann surface.
Communicated by Suayip Yuzbasi.
Received November 22, 2019; January 15, 2020
ON THE ENUMERATION OF THE ROOTS
OF ARBITRARY SEPARABLE EQUATIONS USING
HW HYPER-LAMBERT MAPS
IOANNIS GALIDAKIS
Department of Mathematics
Agricultural University of Athens
Greece
e-mail: jgal@aua.gr
Abstract
In this article we use the HW maps to solve arbitrary equations ,0
=
f by
providing an effective enumeration of the roots of f, as these project on and at
the branches of the HW maps. This is just an enumeration of the projection
points (roots) of a pin-line on the Riemann surface of f through HW.
1. Introduction
The HW maps have been used to determine the attractors of the
infinite exponential whenever it falls into a p-cycle in [3] and in [2] to
solve certain transcendental equations such as Kepler’s equation. They
have also been used in [4] to solve in closed form the generalized Abel
differential equation. Here we display a simple algebraic scheme which
can be used to utilize the solution of arbitrary equations using the HW
maps, by providing an effective enumeration of all the roots of arbitrary
IOANNIS GALIDAKIS
2
equations ,0=f using the branches of the maps HW. Imagine an
arbitrary multivalued f, for which we force
(
)
.0
=
xf We line-pin the
entire Riemann surface of f from top to bottom starting at the complex
origin. The local projection pin points i
z will be exactly the roots of
.0=f Because the branches of HW can be enumerated starting at the
origin, all the roots i
z of 0
=
f can therefore be enumerated and
referenced by approximating just an pin through the origin.
2. Definitions
Suppose
()
zfn are non-vanishing identically complex functions, with
.
0N∈≤ nn We define
(
)
CCN →
×
:zFn as:
Definition 2.1.
()
() ()
>
=
=
−− .1if,
,1if,1
11 ne
n
zF
zFzf
n
nn
Definition 2.2.
()
(
)
.;,,, 121 zFzzfffG nn +
⋅
=
…
If ,0=n then
()
.zzG = If ,1
=
n then
()
(
)
.; 1
1zf
zezfG = If ,2=n
then
()
()
.;,, 1
2
12
zf
ef
zezffG = When we write about the HW, we can use
the terminology
()
,; zG … meaning that the corresponding function
includes meaningful terms-parameters. The order of the functions is
immaterial and we can re-order them to get to the function of interest
here, which is the inverse of
(
)
,; zG … denoted by,
(
)
.;,,,HW 21 yfff n
… (1)
In other words G and HW satisfy the functional relation:
(
)
(
)
yyG
=
;HW; …… (2)
by supposing always that the list of parameters is identical on both sides.
These maps have been called generalized hyper-Lambert HW functions
ON THE ENUMERATION OF THE ROOTS OF … 3
and in general they are multivalued. We note that when
()
yn HW,1
=
satisfies a more general form which comes from the Lambert function W,
i.e.,
()
.
1yze zf = The Lambert function satisfies .yzez= The existence of
all the HW is guaranteed in all cases by the Lagrange Inversion Theorem
(see ([5], p. 201-202)).
3. An Indexing Scheme for the HW Maps
3.1. An algebraic scheme
For the complex maps log and W, their indexing scheme is the
simplest possible, that is
(
)
z,log k and
(
)
.,, Z
∈
kk zW There exists an
indexing scheme which indexes identically the mappings HW but it is not
integral. Dubinov and Galidakis in [1] solve Kepler’s equation, using the
following algebraic inversion:
(
)
⇒
=
⋅− MEE sin
(
)
⇒=
−M
E
E
Esin
1
()
⇒=⋅ ⋅− MeE Eincs1log
(
)
(
)
[
]
.;incs1logHW MxE
⋅
−
=
(3)
The inversion above can be generalized producing a removable pole
at 0
z of multiplicity n. Setting
()
,
0n
zzw −= with 0
z such that
()
,
0yzf = we have:
()
⇒
=
yzf
() ()
()
⇒=
−
⋅− y
zz
zf
zz n
n
0
0
(()
)
⇒=⋅ yew w
zf
log
IOANNIS GALIDAKIS
4
(
)
⇒
=y
w
zf
w;logHW
() ()
()
⇒
−
=− y
zz
zf
zz n
n;logHW
0
0
()
()
.;logHW 0
1
0
zy
zz
zf
z
n
n+
−
= (4)
The scheme above gives an index into the set of the HW functions, in
the form of a functional parameter as
()
()
.log
0
−n
zz
zf Now, if we know
()
,zf this scheme can give identities which must hold identifying this
way the corresponding function.
We can now list how the most important categories of complex
functions are solved based on this index.
3.2. Polynomial functions
Suppose then that
() ( )
.
1
k
k
k
n
N
zzzf −= ∏
=
Keeping
k
fixed and setting
()
,
k
k
n
zzw −= we have
(
)
⇒
=y
w
zf
w;logHW
()
()
⇒+
−
=k
k
k
k
zy
zz
zf
z
n
n
1
;logHW
[
()() ( )
]
.;loglogHW
1
kkk
k
zyzznzfz n+−−= (5)
ON THE ENUMERATION OF THE ROOTS OF … 5
Theorem 3.1. If
() ( )
k
k
k
n
N
zzzf −= ∏
=1
is a complex polynomial function,
then the inverse of
()
zf relative to y is given by the function HW and the
last equation of (5), whose Riemann surface has at most k
k
nm
N
∑
=
=
1
branches, indexed by m, with .N
∈
k
Proof. The last expression of (5) is true for any
{
}
,,,2,1 N…
∈
k
therefore the multiplicity is at least m because for each
k
the
multiplicity is at least k
n and each k
n may give different branches. This
means that the expression can indexfully all the branches of the
corresponding HW using only an integral index .k
Theorem 3.2. If
()
zf is a complex polynomial function, the roots of
()
yzf = are given directly by a suitable HW function.
Proof. Using Equation (1) of Definition 2.2, follows that for each
()
,00;HWHW, =… therefore calculating the corresponding HW of the
last equations in (5) at ,0
=
y forces k
zz
=
and these are the roots of
()
.yzf = Therefore, we can extract all the roots of equation
()
,yzf =
manually. The first root, suppose ,
1
z is extracted as,
(
)
,;logHW
1
=y
z
zf
z
()
(
)
.
1
1zz
yzf
zg −
−
=
IOANNIS GALIDAKIS
6
Having the root ,
1
z the rest of the roots can be extracted recursively for
11 −≤≤ Nk as,
(
)
,;logHWlim
0
1
=+
→
+
z
zg
zk
k
()
(
)
,
1
1+
+−
=
k
k
kzz
zg
zg
and the Theorem follows.
3.3. Rational functions
We suppose that
()
(
)
(
)
,zQzPzf
=
with
(
)
(
)
zQzP , polynomial
functions. We have similar results here.
Theorem 3.3. If
()
(
)
(
)
zQzPzf
=
is a complex rational function
such that
()
(
){}
,deg,degmax QPN = then the inverse of
(
)
zf relative to y
is given by:
() ()
()
,;logHW
1
k
k
k
k
zy
zz
zQyzP
z
n
n+
−
⋅−
=
whose Riemann surface has at most k
k
nm
N
∑
=
=
1
branches, indexed by m,
with .N∈k
Proof. If
() ()
(
)
,zQyzPzF
⋅
−
= then
(
)
zF is a polynomial of degree
N, in which case the Theorem follows similarly, with
(
)
zf replaced by
()
.zF
Theorem 3.4. If
()
zf is a complex rational function, the roots of
()
yzf = can be given by a suitable HW function.
ON THE ENUMERATION OF THE ROOTS OF … 7
Proof. Similarly, if
(
)
(
)
(
)
,zQyzPzF
⋅
−
=
then
(
)
zF is polynomial
map of degree N, therefore we can extract its roots as:
(
)
,;logHWlim
0
1
=+
→
z
zF
z
()
(
)
.
1
1zz
zF
zg −
=
Having ,
1
z the rest of the roots can be extracted recursively for
11 −≤≤ Nk as,
(
)
,;logHWlim
0
1
=+
→
+
z
zg
zk
k
()
(
)
,
1
1+
+−
=
k
k
kzz
zg
zg
and the Theorem follows.
We observe that when
(
)
,1
=
zQ the case of a polynomial function
arises.
3.4. Analytic functions
For an analytic function
() ( )
n
n
n
zzzf 0
0
−⋅α= ∑
∞
=
in some region
,C⊆D with ,
0Dz ∈ we have similar results.
Theorem 3.5. If
() ( )
n
n
n
zzzf 0
0
−⋅α= ∑
∞
=
is a complex analytic function,
then the inverse of
()
zf relative to y is given by a suitable HW function:
()
()
,;logHW
1
k
k
k
k
zy
zz
zf
z
n
n+
−
=
whose Riemann surface has infinitely many branches given by .N
∈
n
IOANNIS GALIDAKIS
8
Proof. Suppose
() ( )
,
0
0
n
n
N
n
NzzzT −⋅α= ∑
=
is the corresponding Taylor
polynomial of degree N. Then
(
)
zTN is obviously a polynomial function,
therefore the inverse of
(
)
zTN relative to y is given again by Theorem
3.1.
()
()
,;logHW
1
k
k
k
k
zy
zz
zT
z
n
n
N+
−
= (6)
() ()
zfzTN→ uniformly in compact subsets and the HW are analytic
([3]), therefore (6) implies that the inverse is given by:
()
()
⇒+
−
=∞→ k
k
k
k
zy
zz
zT
z
n
n
N
N
1
;logHWlim
()
()
⇒+
−
=∞→ k
k
k
k
zy
zz
zT
z
n
n
N
N
1
;
lim
logHW
()
()
,;logHW
1
k
k
k
k
zy
zz
zf
z
n
n+
−
= (7)
and the Theorem follows.
We observe that in this case the inverse function has infinitely many
branches, since N is not bounded.
Theorem 3.6. If
()
zf is a complex analytic function, the roots of
()
yzf = are given again by a HW function.
ON THE ENUMERATION OF THE ROOTS OF … 9
Proof. We can extract the roots as:
(
)
,;logHW
1
=y
z
zf
z
()
(
)
.
1
1zz
yzf
zg −
−
= (8)
The rest of the roots can be again extracted recursively for
k
≤
1 as,
(
)
,;logHWlim
0
1
=+
→
+
z
zg
zk
k
()
(
)
,
1
1+
+−
=
k
k
kzz
zg
zg (9)
and the Theorem follows.
4. HW Functional Index
An open problem set in ([2], p. 1114-1115) is whether there is a way
to effectively index the numbering of the branches of the HW functions.
With the following theorem we show that the answer is affirmative.
Theorem 4.1. If
()
zf is a complex function and ,, NC
∈
∈
k
k
z such
that
()
yzf =
k and suppose
(
)
zgk follows as in Equations (8)-(9). Then, if
HW is the inverse of
()
zf relative to y, the following scheme covers all the
branches of this inverse of
(
)
:zf
()
(
)
()
>
=
=
+
→
+
.0,;log,HWlim
,0,;log,0HW
0
1
kk
k
k
k
if
z
zg
ify
z
zf
z
Proof. The proof follows from (3) along with Theorems 3.1, 3.3, and
3.5. Note that for a specific analytic f expanded around ,
k
z we define
() ()
()
.log
−
=
k
k
n
zz
zf
zF The map F creates a Laurent series with residue
IOANNIS GALIDAKIS
10
()
,exp
k
n
a which is gotten from the HW through the Residue Theorem of
Cauchy for f, with winding number k
n
a around .
k
z Consequently, the
repeated application of
()
k
gF via for ,
k
zz
=
extracts recursively all the
roots k
z of the inverse and as such it can be used as an index for the
corresponding Riemann surface.
5. Conclusion
The HW maps can solve any equation ,0
=
f provided it can be
brought into a separable form with all z’s on the left and one w on the
right. Further, the enumeration of the roots is origin consistent relative
to the force of f.
References
[1] A. E. Dubinov and I. N. Galidakis, Explicit solution of the Kepler equation, Physics
of Particles and Nuclei Letters 4(3) (2007), 213-216.
DOI: https://doi.org/10.1134/S1547477107030028
[2] I. N. Galidakis, On some applications of the generalized hyper-Lambert functions,
Complex Variables and Elliptic Equations 52(12) (2007), 1101-1119.
DOI: https://doi.org/10.1080/17476930701589563
[3] I. N. Galidakis, On solving the p-th complex auxiliary equation
()
()
,zzf p= Complex
Variables, Theory and Application: An International Journal 50(13) (2005), 977-997.
DOI: https://doi.org/10.1080/02781070500156827
[4] P. Nastou, Y. Stamatiou and A. Tsiakalos, Solving a class of odes arising in the
analysis of a computer security process using generalized hyper-Lambert functions,
International Journal of Applied Mathematics and Computation (IJAMC) 4(3)
(2012), 67-76.
[5] S. Saks and A. Sygmund, Analytic Functions, Hafner Publishing Company, New
York, 1952.
g
ON THE ENUMERATION OF THE ROOTS OF … 11
Appendix: Programming with the HW Maps
Code for the HW maps is given below. Arguments are HW (functional
index, y, n)
restart;
Digits:=40;
HW := proc ()
local y, n, c, s, p, sol, i, aprx, dy, dist, r, newr, oldr,
fun, dfun, eps;
if nargs < 2 then ERROR(“At least two arguments required”) end
if;
n := args[-1]; y := args[-2]; c := [args[1 .. -3]];
if y = 0 then 0 else dist := infinity;
eps:=1e-10; fun := 1; for i from 1 to nargs-2 do fun :=
exp(c[-i]*fun) end do;
fun := z*fun-y; dfun := diff(fun, z);
s := series(fun, z, n); p := convert(s, polynom);
sol := {fsolve(p = 0, z, complex)};
for i from 1 to nops(sol) do aprx := evalf(subs(z = op(i,
sol), fun));
dy := evalf(abs(aprx)); if dy <= dist then r := op(i, sol);
dist := dy end if end do; oldr := r;
newr := r-evalf(subs(z = r, fun)/subs(z = r, dfun));
for i from 1 to 1000 while abs((oldr-newr)/oldr)>eps do oldr
:= newr;
newr := newr-evalf(subs(z = newr, fun)/subs(z = newr, dfun))
end do;
newr end if end proc:
IOANNIS GALIDAKIS
12
Example 1. Using the program with five decimal digits accuracy to
solve the equation
()()
(
)
,2532
=
−
−
−zzz
y:=2;
f:=z->(z-2)*(z-3) * (z-5);
z1:=HW(log(f(z)/z),y,10);
g1:=z->(f(z)-y)/(z-z1);
z2:=HW(log(g1(z)/z),1e-20,10);
g2:=z->g1(z)/(z-z2);
z3:=HW(log(g2(z)/z),1e-20,10);
gives:
iz 69160.036523.2
1
−
iz 69160.036523.2
2
+
26953.5
3z
Using the program for an approximate solution with Maple,
solve(f(z)=y,z);
evalf(%);
gives:
5.26953, 2.36523 + 0.69160i, 2.36523 − 0.69160i.
Example 2. Using the program to five digits of accuracy to solve the
equation
()()
(
)
(
)
,21532
=
−
−
−− zzzz
y:=2;
f:=z->(z-2)*(z-3)/(z-5)/(z-1);
P:=unapply(numer(f(z)),z);
Q:=unapply(denom(f(z)),z);
ON THE ENUMERATION OF THE ROOTS OF … 13
F:=unapply(P(z)-y*Q(z),z);
z1:=HW(log(F(z)/z),1e-10,10);
g1:=z->F(z)/(z-z1);
z2:=HW(log(g1(z)/z),1e-10,10);
gives:
37228.6
1z
62771.0
2z
Using Maple approximation code,
solve(F(z)=y,z);
evalf(%);
gives:
6.37228, 0.62771.
Example 3. Using the program to five decimals of accuracy to solve
the equation
()
,21sin =z
y:=1/2;
f:=z->sin(z);
z1:=HW(log(f(z)/z),y,10);
g1:=z->(f(z)-y)/(z-z1);
z2:=HW(log(g1(z)/z),1e-20,10);
g2:=z->g1(z)/(z-z2);
z3:=HW(log(g2(z)/z),1e-20,10);
IOANNIS GALIDAKIS
14
gives:
52359.0
1z
61799.2
2z
66519.3
3
−
z
…
4
z
The results are approximations of the numbers ,,67,65,6 …
π
−
π
π
which are the roots of
(
)
.21sin
=
z Many more complex equations can
be solved here, provided they are separable and the terms are analytic,
like.
Example 4. Using the code with five decimal accuracy to solve the
equation
() ()() ()
,21tanh1sinexpsin =++ zzz
y=1/2;
f:=sin(z)+exp(sin(z))/sqrt(1+tanh(z));
z1:=HW(log(f(z)/z), y, 10);
g1:= z->(f(z)-y)/(z-z1);
z2:=HW(log(f1(z)/z),1e-20,10);
g2:=z->g1(z)/(z-z2);
z3:=HW(log(g2(z)/z),1e-20,10);
g3:=z->g2(z)/(z-z3);
z4:=HW(log(g3(z)/z), 0.1e-19, 10);
gives:
37435.0
1
−
z
71811.1
2
−
z
26659.3
3z
15846.6
4z
ON THE ENUMERATION OF THE ROOTS OF … 15
While using Maple approximation code,
solve(f(z)=y,z);
gives an open answer in terms of “RootOf”, i.e., it cannot relay the
roots directly.
Example 5. Using the code with five decimal accuracy to solve the
equation ,254 23 =+− zzz
y:=2;
f:=z->z^3-4*z^2+5*z;
z1:=HW(log(f(z)/z),y,10);
g1:=z->(f(z)-y)/(z-z1);
z2:=HW(log(f1(z)/z),1e-20,10);
g2:=z->g1(z)/(z-z2);
z3:=HW(log(g2(z)/z),1e-20,10);
gives:
2
1z
00005.1
2z
00000.1
3z
The description calculates correctly roots with multiplicity greater
than 1.
The example is
() ( )( )
,21 2−−=− zzyzf therefore the multiplicity of
the root 1 is indeed 2.
