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Applied Mathematics and Computation 397 (2021) 125954
Contents lists available at ScienceDirect
Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
A multivalued logarithm on time scales
Douglas R. Anderson a , ∗, Martin Bohner b
a Department of Mathematics, Concordia College Moorhead,
Minnesota 56562, USA b Department of Mathematics and Statistics,
Missouri University of Science and Technology, Rolla, Missouri
65409-0020, USA
a r t i c l e i n f o
Article history:
Received 16 February 2020
Revised 21 December 2020
Accepted 28 December 2020
MSC:
34N05
Keywords:
Dynamic equations
Cylinder transformation
Logarithm
Time scales
Cayley transformation
a b s t r a c t
A new definition of a multivalued logarithm on time scales is
introduced for delta-
differentiable functions that never vanish. This new logarithm
arises naturally from the
definition of the cylinder transformation that is also the
wellspring of the definition of
exponential functions on time scales. This definition will lead
to a logarithm function on
arbitrary time scales with familiar and useful properties that
previous definitions in the
literature lacked.
© 2021 Elsevier Inc. All rights reserved.
1. Introduction
Recall that a time scale T is a closed subset of the real line.
This induces the forward jump operator σ : T → T viaσ (t) = inf { s
∈ T : s > t} , and the graininess function μ(t) := σ (t) − t;
see [1–4] for more details. A recurring open problemfor time scales
and dynamic equations [3,4,9] has been the following [5] : On time
scales, define and present the properties
of a “nice” logarithm function. The aim of what follows below is
to introduce on time scales a novel multivalued logarithm
arising from the cylinder transformation employed in definitions
of exponential functions for dynamic equations.
The development of this logarithm on general time scales will
proceed as follows. In Section 2 , we extend the definition
of the traditional single valued cylinder transformation to a
multivalued cylinder transformation. This transformation has
useful properties across the circle plus (�) and circle dot (�)
operations, and is the basis for the definition of the log-
arithm, for non-vanishing delta-differentiable functions. In
Section 3 , nice properties of this new logarithm are shown to
hold. Section 4 establishes a similar logarithm for the nabla
case. In Section 5 , the Cayley cylinder transformation is also
considered, and is shown to lead to the very same logarithm. In
Section 6 , we give a listing of extant logarithm functions on
time scales from the literature. Finally, in Section 7 , we give
a numerical comparison of the various logarithms on a specific
time scale, and give numerous examples on various time scales
illustrating the properties of the new one. For trends on
time scales generally, see the recent works [1,2,7,8] .
∗ Corresponding author. E-mail addresses: [email protected]
(D.R. Anderson), [email protected] (M. Bohner).
https://doi.org/10.1016/j.amc.2021.125954
0 096-30 03/© 2021 Elsevier Inc. All rights reserved.
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D.R. Anderson and M. Bohner Applied Mathematics and Computation
397 (2021) 125954
2. A new logarithm on time scales
We begin our presentation of a new definition on general time
scales of a logarithm for dynamic equations, with some
motivation provided by the definition of exponential functions
for dynamic equations based on the cylinder transformation.
The following definition [3, Definition 2.21] (see also Hilger
[9, Section 7] ) is the original cylinder transformation; a
modified
cylinder transformation will also be examined, in Section 5
.
Definition 2.1 (Single Valued Cylinder Transformation) . Fix h
> 0 , and define the cylinder transformation ξh : C h → Z h
by
ξh (z) = {
1
h Log (1 + zh ) for h � = 0
z for h = 0 , (2.1)
where C is the set of complex numbers,
C h = {
z ∈ C : z � = −1 h
} , Z h =
{ z ∈ C : −π
h < Im (z) ≤ π
h
} , (2.2)
and Log represents the principal complex logarithm function.
The following definition is [3, Definition 2.25] .
Definition 2.2 (Regressive Function) . A function p : T → R is
regressive granted 1 + μ(τ ) p(τ ) � = 0 for each τ ∈ T κ
holds. We will denote via R the set of all rd-continuous and
regressive functions p : T → R . The following definition is [3,
Definition 2.30] .
Definition 2.3 (Exponential Function) . For functions p ∈ R ,
the time scales exponential function is formulated via
e p (t, s ) = exp (∫ t
s
ξμ(τ ) (p(τ ))�τ
)for s, t ∈ T ;
here, ξh (z) is the cylinder transformation given in (2.1) .
We now set the foundation for offering a new definition of
logarithms on time scales. This definition will be of a multi-
valued function, for which we need to modify the single valued
cylinder function given in (2.1) .
Definition 2.4 (Multivalued Cylinder Transformation) . Fix h
> 0 , and define the multivalued cylinder transformation ζh : C
h → C by
ζh (z) = {
1
h log (1 + zh ) for h � = 0
z for h = 0 , (2.3)
where the set of complex numbers is C , the set C h is given in
(2.2) , and log represents the multivalued complex logarithm
function.
Lemma 2.5. Let f, g : T → C be �-differentiable functions with
f, g � = 0 on T , and let the multivalued cylinder transformation
ζbe given by (2.3) . Then, for fixed τ ∈ T κ ,
ζμ(τ )
((f �
f �
g �
g
)(τ )
)= ζμ(τ )
(f �(τ )
f (τ )
)+ ζμ(τ )
(g �(τ )
g(τ )
).
Proof. First, note that the useful yet simple formula f σ = μ f
� + f (suppressing the variable) implies ( f g) �
f g = f
σ g � + f �g f g
= ( f + μ f �) g �
f g + f
�
f
= f �
f + g
�
g + μ f
�g �
f g
= f �
f �
g �
g .
From this, we observe that for fixed τ ∈ T κ ,
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D.R. Anderson and M. Bohner Applied Mathematics and Computation
397 (2021) 125954
ζμ(τ )
((f �
f �
g �
g
)(τ )
)= ζμ(τ )
(( f g) �(τ )
( f g)(τ )
)
=
⎧ ⎨ ⎩ 1
μ(τ ) log
(1 + μ(τ ) ( f g)
�(τ )
( f g)(τ )
)for μ(τ ) � = 0
( f g) �(τ ) ( f g)(τ )
for μ(τ ) = 0
=
⎧ ⎪ ⎨ ⎪ ⎩ 1
μ(τ ) log
(( f g) σ (τ )
( f g)(τ )
)for μ(τ ) � = 0 (
f �
f �
g �
g
)(τ ) for μ(τ ) = 0
=
⎧ ⎪ ⎨ ⎪ ⎩ 1
μ(τ ) log
(f σ (τ )
f (τ )
)+ 1
μ(τ ) log
(g σ (τ )
g(τ )
)for μ(τ ) � = 0 (
f �
f + g �
g
)(τ ) for μ(τ ) = 0
=
⎧ ⎨ ⎩ 1
μ(τ ) log
(( f + μ f �)(τ )
f (τ )
)+ 1
μ(τ ) log
((g + μg �)(τ )
g(τ )
)for μ(τ ) � = 0
f �(τ ) f (τ )
+ g �(τ ) g(τ ) for μ(τ ) = 0
= ζμ(τ ) (
f �(τ )
f (τ )
)+ ζμ(τ )
(g �(τ )
g(τ )
).
The proof is complete. �
Lemma 2.6. Let α ∈ R , and let p : T → C be a �-differentiable
function with p � = 0 on T . For the multivalued cylinder
transfor-mation ζ given by (2.3) and for fixed τ ∈ T κ ,
ζμ(τ )
((α �
p �
p
)(τ )
)= αζμ(τ )
(p �(τ )
p(τ )
).
Proof. Let α ∈ R , and let p : T → C be a �-differentiable
function with p � = 0 on T . Then [4, Theorem 2.43] yields 1 + μ(α
� f ) = (1 + μ f ) α
on T κ for f = p �p . It follows that for fixed τ ∈ T κ ,
ζμ(τ )
((α �
p �
p
)(τ )
)
=
⎧ ⎪ ⎨ ⎪ ⎩ 1
μ(τ ) log
(1 + μ(τ )
(α �
p �
p
)(τ )
)for μ(τ ) � = 0 (
α � p �
p
)(τ ) for μ(τ ) = 0
=
⎧ ⎨ ⎩ 1
μ(τ ) log
(1 + μ(τ ) p
�(τ )
p(τ )
)αfor μ(τ ) � = 0
α p �(τ ) p(τ ) for μ(τ ) = 0
= α
⎧ ⎨ ⎩ 1
μ(τ ) log
(1 + μ(τ ) p
�(τ )
p(τ )
)for μ(τ ) � = 0
p �(τ ) p(τ ) for μ(τ ) = 0
= αζμ(τ ) (
p �(τ )
p(τ )
).
The proof is complete. �
Definition 2.7 (Logarithm Function) . Given a �-differentiable
function p : T → C with p � = 0 on T , the multivalued loga-rithm
function on time scales is given by
p (t, s ) = ∫ t
s
ζμ(τ )
(p �(τ )
p(τ )
)�τ for s, t ∈ T ,
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D.R. Anderson and M. Bohner Applied Mathematics and Computation
397 (2021) 125954
(
(i
where ζh (z) is the multivalued cylinder transformation given in
(2.3) . Define the principal logarithm on time scales to be
L p (t, s ) = ∫ t
s
ξμ(τ )
(p �(τ )
p(τ )
)�τ for s, t ∈ T ,
where ξh (z) is the single valued cylinder transformation given
in (2.1) .
Remark 2.8. According to this definition, if p ≡ constant, then
p (t, s ) = 0 for each t, s ∈ T . Thus, this logarithm does
notdistinguish between either constants or constant multiples of
functions. We moreover note here that even when we restrict
the time scale to T = R , the dynamics along the negative and
positive real line necessitate the existence of a logarithm
withprincipal and multiple values, making a multivalued logarithm
on general time scales both natural and expected, though
heretofore unexplored.
3. Properties of the logarithm
Using the definition of the multivalued logarithm on time scales
given above in Definition 2.7 , we establish the following
properties.
Theorem 3.1. If p : T → C is a �-differentiable function with p
� = 0 on T , then exp ( L p (t, s ) ) = e p �
p
(t , s ) , t , s ∈ T .
In particular, if p ∈ R , then exp
(L e p (t, s )
)= e p (t , s ) , t , s ∈ T .
Proof. Presuming p : T → C is a �-differentiable function with p
� = 0 on T ,
L p (t, s ) = ∫ t
s
ξμ(τ )
(p �(τ )
p(τ )
)�τ.
Now, exponentiate both sides and use the definition of e p (t, s
) , the exponential function. �
Theorem 3.2 (Logarithm of Product, Quotient, & Power) .
Presume f, g, p : T → C are �-differentiable functions with f, g, p
� = 0on T . Then, for s, t ∈ T and α ∈ R , we have the following:
(i) f g (t, s ) = f (t, s ) + g (t, s ) , ii) f
g
(t, s ) = f (t, s ) − g (t, s ) , ii) p α (t, s ) = α p (t, s )
. Proof. Presume f, g, p : T → C are �-differentiable functions
with f, g, p � = 0 on T . Then, for s, t ∈ T , we have viaLemma 2.5
and its proof that
f g (t, s ) = ∫ t
s
ζμ(τ )
(( f g) �(τ )
( f g)(τ )
)�τ
= ∫ t
s
ζμ(τ )
((f �
f �
g �
g
)(τ )
)�τ
= ∫ t
s
ζμ
(f �(τ )
f (τ )
)�τ +
∫ t s
ζμ
(g �(τ )
g(τ )
)�τ
= f (t, s ) + g (t, s ) . In a similar manner,
f g
(t, s ) = ∫ t
s
ζμ(τ )
( (f g
)�(τ ) (
f g
)(τ )
) �τ
= ∫ t
s
ζμ(τ )
((f �
f �
g �
g
)(τ )
)�τ
= ∫ t
s
ζμ
(f �(τ )
f (τ )
)�τ −
∫ t s
ζμ
(g �(τ )
g(τ )
)�τ
= f (t, s ) − g (t, s ) . Let α ∈ R . For the multivalued
cylinder transformation ζ given by (2.3) and for fixed τ ∈ T κ
,
ζμ(τ )
((α �
p �
p
)(τ )
)= αζμ(τ )
(p �(τ )
p(τ )
)
4
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D.R. Anderson and M. Bohner Applied Mathematics and Computation
397 (2021) 125954
using Lemma 2.6 . Moreover, by [4, Theorem 2.37] , we have
( p α) �
p α= α � p
�
p .
Consequently,
p α (t, s ) = ∫ t
s
ζμ(τ )
(( p α)
�(τ )
p α(τ )
)�τ
= ∫ t
s
ζμ(τ )
((α �
p �
p
)(τ )
)�τ
= ∫ t
s
αζμ(τ )
(p �(τ )
p(τ )
)�τ
= α p (t, s ) . The argument proves sufficient. �
Theorem 3.3. Let p : T → R be a �-differentiable function with p
� = 0 on T . Then, for s, t ∈ T , we have
�p (t, s ) = {
1 μ(t) log
(p σ (t) p(t)
)for μ(t) � = 0
p �(t) p(t)
for μ(t) = 0 , where �-differentiation is with respect to t.
Proof. Using the definition of the logarithm and
�-differentiating with respect to t,
�p (t, s ) = ζμ(t) (
p �(t)
p(t)
)
=
⎧ ⎨ ⎩ 1
μ(t) log
(1 + μ(t) p
�(t)
p(t)
)for μ(t) � = 0
p �(t) p(t)
for μ(t) = 0 .
Now substitute μp � = p σ − p. The argument proves sufficient.
�
4. The nabla case
A logarithm is also possible for the nabla case.
Definition 4.1 (Cylinder Transformation) . For h > 0 , define
the single valued cylinder transformation ̂ ξh : ̂ C h → Z h by ̂
ξh (z) =
{ −1 h
Log (1 − zh ) for h � = 0 z for h = 0
(4.1)
and the multivalued cylinder transformation ̂ ζh : ̂ C h → C by
̂ ζh (z) =
{ −1 h
log (1 − zh ) for h � = 0 z for h = 0 .
(4.2)
Here C is the set of complex numbers, Z h is in (2.2) ,
̂ C h = { z ∈ C : z � = 1 h } , and as before, Log represents
the principal complex logarithm function.
The following definition is [4, Definition 3.4] .
Definition 4.2 (Regressive Function) . A function p : T → R is
ν-regressive granted 1 − ν(t) p(t) � = 0 for each t ∈ T κ
holds. Let ̂ R signify the set of all ld-continuous and
ν-regressive functions p : T → R . The following definition is [4,
Definition 3.10] .
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D.R. Anderson and M. Bohner Applied Mathematics and Computation
397 (2021) 125954
Definition 4.3 (Exponential Function) . Let t, s ∈ T . For
functions p ∈ ̂ R , the time scales nabla exponential function is
formu-lated via
̂ e p (t, s ) = exp (∫ t s
̂ ξν(τ ) (p(τ )) ∇τ), where ̂ ξh (z) is the single valued
cylinder transformation given in (4.1) .
We now offer a new definition of logarithms for the nabla case
on time scales.
Definition 4.4 (Logarithm Function) . Given a ∇-differentiable
function p : T → R with p � = 0 on T , the multivalued
nablalogarithm function on time scales is given by
̂ p (t, s ) = ∫ t s
̂ ζν(τ ) ( p ∇ (τ ) p(τ )
)∇τ for s, t ∈ T ,
where ̂ ζh (z) is the multivalued cylinder transformation given
in (4.2) , while the principal nabla logarithm is given by ̂ L p
(t, s ) = ∫ t
s
̂ ξν(τ ) ( p ∇ (τ ) p(τ ) )
∇τ for s, t ∈ T ,
where ̂ ξh (z) is the single valued nabla cylinder
transformation given in (4.1) Properties analogous to those given
earlier can be established for the nabla case as well.
5. Logarithms for Cayley-exponential functions
In [6] , the author introduced another time scales exponential
function, dubbed the Cayley-exponential function, defined
by
E p (t, s ) = exp (∫ t
s
�μ(τ ) (p(τ ))�τ
), (5.1)
where p : T → C is rd-continuous and satisfies the regressivity
condition μ(τ ) p(τ ) � = ±2 for all τ ∈ T κ , and the
modifiedcylinder transformation � is given by
�h (z) = 1
h Log
(1 + 1
2 zh
1 − 1 2
zh
), �0 (z) = z, (5.2)
for h > 0 . Once more, Log represents the principal complex
logarithm. Consider the multivalued function version of (5.2)
de-
noted, i.e.,
ψ h (z) = 1
h log
(1 + 1
2 zh
1 − 1 2
zh
), ψ 0 (z) = z, (5.3)
where log represents the multivalued complex logarithm. We
introduce the following Cayley-logarithm functions on time
scales.
Definition 5.1. For a �-differentiable function p : T → C with p
� = 0 on T , the multivalued Cayley-logarithm function ontime
scales is given by
caylog p (t, s ) = ∫ t
s
ψ μ(τ )
(2 p �(τ )
p(τ ) + p σ (τ )
)�τ for s, t ∈ T ,
where ψ h (z) is the multivalued cylinder transformation given
in (5.3) . Define the principal Cayley-logarithm on time scalesto
be
CayLog p (t, s ) = ∫ t
s
�μ(τ )
(2 p �(τ )
p(τ ) + p σ (τ )
)�τ for s, t ∈ T ,
where �h (z) is the single valued cylinder transformation given
in (5.2) .
Lemma 5.2. The Cayley-logarithm functions are well-defined
functions.
Proof. For a �-differentiable function p : T → C with p � = 0 on
T , we need to show that
μ(τ ) 2 p �(τ )
p(τ ) + p σ (τ ) � = ±2 ,
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D.R. Anderson and M. Bohner Applied Mathematics and Computation
397 (2021) 125954
in other words, that the regressivity condition holds. The
following are equivalent:
2 μ(τ ) p �(τ )
p(τ ) + p σ (τ ) = ±2 ⇐⇒ p σ (τ ) − p(τ ) p(τ ) + p σ (τ ) =
±1
p σ (τ ) − p(τ ) = ±( p(τ ) + p σ (τ ) ) ⇐⇒ p σ (τ ) ∓ p σ (τ )
= p(τ ) ± p(τ ) , so that we have either 0 = 2 p(τ ) or 2 p σ (τ )
= 0 , both contradictions. �
Theorem 5.3. For a �-differentiable function p : T → C with p �
= 0 on T ,
caylog p (t, s ) = p (t , s ) and CayLog p (t , s ) = L p (t, s
) (5.4)for all s, t ∈ T .
Proof. Consider (5.3) . For fixed τ ∈ T κ with μ(τ ) � = 0 ,
observe that
ψ μ(τ )
(2 p �(τ )
p(τ ) + p σ (τ )
)= 1
μ(τ ) log
( 1 + 1
2 2 p �(τ )
p(τ )+ p σ (τ ) μ(τ )
1 − 1 2
2 p �(τ ) p(τ )+ p σ (τ ) μ(τ )
)
= 1 μ(τ )
log
( 1 + μ(τ ) p �(τ )
p(τ )+ p σ (τ ) 1 − μ(τ ) p �(τ )
p(τ )+ p σ (τ )
)
= 1 μ(τ )
log
( 1 + p σ (τ ) −p(τ )
p(τ )+ p σ (τ ) 1 − p σ (τ ) −p(τ )
p(τ )+ p σ (τ )
)
= 1 μ(τ )
log
(p σ (τ )
p(τ )
)= 1
μ(τ ) log
(p(τ ) + μ(τ ) p �(τ )
p(τ )
)= ζμ(τ )
(p �(τ )
p(τ )
)for ζh defined in (2.3) . For fixed τ ∈ T κ with μ(τ ) = 0 , we
have τ = σ (τ ) and
2 p �(τ )
p(τ ) + p σ (τ ) = p �(τ )
p(τ ) .
Consequently,
ψ μ(τ )
(2 p �(τ )
p(τ ) + p σ (τ )
)= 2 p
�(τ )
p(τ ) + p σ (τ ) = p �(τ )
p(τ ) = ζμ(τ )
(p �(τ )
p(τ )
).
Thus, in either case, we have
ψ μ(τ )
(2 p �(τ )
p(τ ) + p σ (τ )
)= ζμ(τ )
(p �(τ )
p(τ )
).
It follows that
caylog p (t, s ) = ∫ t
s
ψ μ(τ )
(2 p �(τ )
p(τ ) + p σ (τ )
)�τ =
∫ t s
ζμ(τ )
(p �(τ )
p(τ )
)�τ = p (t, s ) .
Similarly, we have
CayLog p (t, s ) = L p (t, s ) , completing the proof. �
Remark 5.4. The previous theorem and proof may be generalized,
as we will now show. Let θ ∈ [0 , 1] , and set
ψ θh (z) = 1
h log
(1 + (1 − θ ) zh
1 − θzh
), ψ θ0 (z) = z. (5.5)
Then, for a �-differentiable function p : T → C with p � = 0 on
T , and for all τ ∈ T κ , we have
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D.R. Anderson and M. Bohner Applied Mathematics and Computation
397 (2021) 125954
ψ θμ(τ )
(p �(τ )
(1 − θ ) p(τ ) + θ p σ (τ )
)= 1
μ(τ ) log
( 1 + (1 − θ ) μ(τ ) p �(τ )
(1 −θ ) p(τ )+ θ p σ (τ ) 1 − θμ(τ ) p �(τ )
(1 −θ ) p(τ )+ θ p σ (τ )
)
= 1 μ(τ )
log
((1 − θ ) p(τ ) + θ p σ (τ ) + (1 − θ ) μ(τ ) p �(τ )
(1 − θ ) p(τ ) + θ p σ (τ ) − θμ(τ ) p �(τ )
)= 1
μ(τ ) log
((1 − θ ) p(τ ) + θ p σ (τ ) + (1 − θ )(p σ (τ ) − p(τ ))
(1 − θ ) p(τ ) + θ p σ (τ ) − θ (p σ (τ ) − p(τ ))
)= 1
μ(τ ) log
(p σ (τ )
p(τ )
)= 1
μ(τ ) log
(p(τ ) + μ(τ ) p �(τ )
p(τ )
)= ζμ(τ )
(p �(τ )
p(τ )
)for ζh defined in (2.3) . For fixed τ ∈ T κ with μ(τ ) = 0 , we
have τ = σ (τ ) and
p �(τ )
(1 − θ ) p(τ ) + θ p σ (τ ) = p �(τ )
p(τ ) .
As a result,
ψ θ0
(p �(τ )
(1 − θ ) p(τ ) + θ p σ (τ )
)= p
�(τ )
(1 − θ ) p(τ ) + θ p σ (τ ) = p �(τ )
p(τ ) = ζ0
(p �(τ )
p(τ )
).
Thus, in either case, we have
ψ θμ(τ )
(p �(τ )
(1 − θ ) p(τ ) + θ p σ (τ )
)= ζμ(τ )
(p �(τ )
p(τ )
)for all θ ∈ [0 , 1] . Consequently,
log θp (t, s ) :=
∫ t s
ψ θμ(τ )
(p �(τ )
(1 − θ ) p(τ ) + θ p σ (τ )
)�τ =
∫ t s
ζμ(τ )
(p �(τ )
p(τ )
)�τ = p (t, s ) .
This ends the remark.
6. Previous logarithms on time scales
As shown in previous sections, the key to arriving at useful
logarithm properties is to allow for a multivalued logarithm,
as exists for the T = R case. Here, we present the previous
definitions of a logarithm on time scales, noting that they are
allsingle valued functions. Moreover, only Definition 2.7 leads to
results as given in Theorem 3.1, Theorem 3.2, Theorem 5.3 , and
Remark 5.4 , justifying this new approach, and emphasizing the
advantages of having a function satisfying familiar properties,
while ensconced in the more general time scales context.
The first logarithm on time scales [10] interprets the integral
∫ t t 0
2
τ + σ (τ ) �τ
as a time scales analogue of ln t . This is understandable,
because if T = R , then τ = σ (τ ) , and ∫ t t 0
2
τ + σ (τ ) �τ = ∫ t
t 0
2
2 τdτ = ln t − ln t 0 .
A recent paper [11] applies iterates of this logarithm to
Riemann–Weber-type equations.
A second approach [5, Section 3] is to view the slightly
different integral ∫ t t 0
1
τ + 2 μ(τ ) �τ
as the time scales version of ln t, due to the same fact that it
reduces to ln t − ln t 0 on T = R , and as it is part of a
solutionform to a certain EulerCauchy dynamic equation whose
differential equation analogue involves the natural logarithm.
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D.R. Anderson and M. Bohner Applied Mathematics and Computation
397 (2021) 125954
A third approach [5, Section 4] could be to define a logarithm
via
L p (t, t 0 ) = ∫ t
t 0
p �(τ )
p(τ ) �τ
for �-differentiable functions p : T → R . Clearly if p(τ ) = τ,
then this is
L p (t, t 0 ) = ∫ t
t 0
p �(τ )
p(τ ) �τ =
∫ t t 0
1
τ�τ,
a form that is similar to its continuous analogue for T = R . A
fourth approach [12] is to take the logarithm to be given by
log T
p(t) = p �(t)
p(t)
for �-differentiable functions p : T → R , where the time scale
logarithm on R does not play the role of the logarithm,clearly, but
rather its derivative. The motivation here is to maintain some
attractive algebraic properties of logarithms, and
to serve in some sense as an inverse to the exponential
function.
A fifth approach [13] , only for time scales such that 1 ∈ T ,
is to define the natural logarithm via
L T (t) = ∫ t
1
1
τ�τ,
which hearkens back to [5, Section 4] . Here the motivation is
clearly that
L R (t) = ln t, L T (1) = 0 , L �T (t) = 1
t .
7. Numerical comparisons and examples of logarithms
Each of the definitions given in the previous section has
advantages and drawbacks, and each one satisfies some of
what one might wish for in a logarithm function. As shown
earlier in this work, however, a multivalued logarithm on time
scales with a definition based on cylinder transformations is a
natural move that leads to nice properties, and has not been
introduced until now. We now consider the following
examples.
Example 7.1. In this example, we compare the values of the
various logarithms on the time scale
T := (−∞ , −k ] ∪ {−k + 1 , −k + 2 , . . . , −1 , 0 , 1 , . . .
, k − 2 , k − 1 } ∪ [ k, ∞ ) , k ∈ N . For p(t) = t on [1 , k + 3]
T , we have the following plot and table of comparison for the
logarithms on time scales mentionedin the literature to date.
Citation Logarithm Value at t = 6 Fig. 1 Color
[10]
k −1 ∑ j=1
2
2 j + 1 + ln (
t
k
)1.75692 blue
[5, Section 3]
k −1 ∑ j=1
1
j + 2 + ln (
t
k
)1.13232 orange
[5, Section 4]
k −1 ∑ j=1
1
j + ln
(t
k
)2.26565 green
[12]
k −1 ∑ j=1
1
j + ln
(t
k
)2.26565 green
[13]
k −1 ∑ j=1
1
j + ln
(t
k
)2.26565 green
Definition 2.7
k −1 ∑ j=1
ln
(j + 1
j
)+ ln
(t
k
)1.79176 red
As can be seen in the table, the new definition presented in
this paper, Definition 2.7 , leads to a unique and accurate
value for this time scale. The comparison of graphs on [1 , 8] T
= { 1 , 2 , 3 , 4 } ∪ [5 , 8] is given in Fig. 1 . In the rest of
this section, we provide numerous examples of the new logarithm
from Definition 2.7 , for various time
scales.
Example 7.2. For T = R ,
p (t, s ) = ∫ t
s
ζμ(τ )
(p �(τ )
p(τ )
)�τ =
∫ t s
p ′ (τ ) p(τ )
d τ = log (
p(t)
p(s )
),
9
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D.R. Anderson and M. Bohner Applied Mathematics and Computation
397 (2021) 125954
Fig. 1. Comparison plot of various logarithms on [1 , k + 3] T
for k = 5 .
where log represents the multivalued complex logarithm function.
For T = h Z ,
��(τ ) = �h �(τ ) := �(h + τ ) − �(τ )
h
and
p (t, s ) = ∫ t
s
ζμ(τ )
(p �(τ )
p(τ )
)�τ =
t h −1 ∑
j= s h
ζh
(�h p( jh )
p( jh )
)h =
t h −1 ∑
j= s h
1
h log
(1 + h �h p( jh )
p( jh )
)h
= t h −1 ∑
j= s h
log
(p( jh + h )
p( jh )
)= log
( t h −1 ∏
j= s h
p(( j + 1) h ) p( jh )
) = log
(p(t)
p(s )
).
For T = q N 0 ,
f �(τ ) = D q f (τ ) := f (qτ ) − f (τ ) (q − 1) τ
and
p (t, s ) = ∫ t
s
ζμ(τ )
(p �(τ )
p(τ )
)�τ =
∑ τ∈ [ s,t)
ζ(q −1) τ
(p �(τ )
p(τ )
)(q − 1) τ
= ∑
τ∈ [ s,t)
1
(q − 1) τ log (
1 + (q − 1) τ p �(τ )
p(τ )
)(q − 1) τ
= ∑
τ∈ [ s,t) log
(p(qτ )
p(τ )
)= log
(p(t)
p(s )
).
This ends the example.
Example 7.3. For real numbers a, b, c, d with a < b < c
< d, set T = [ a, b] ∪ [ c, d] . Assume p : T → C is
differentiable withp � = 0 on T . If s, t ∈ [ a, b) or s, t ∈ [ c,
d] , then μ(τ ) ≡ 0 for τ ∈ [ s, t] , so that by the definition of
the multivalued cylinderfunction (2.3) ,
p (t, s ) = ∫ t
s
p ′ (τ ) p(τ )
d τ = log (
p(t)
p(s )
).
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D.R. Anderson and M. Bohner Applied Mathematics and Computation
397 (2021) 125954
Presume without loss of generality that s ∈ [ a, b] and t ∈ [ c,
d] . Then c = σ (b) , and
p (t, s ) = ∫ t
s
ζμ(τ )
(p �(τ )
p(τ )
)�τ
= (∫ b
s
+ ∫ σ (b)
b
+ ∫ t σ (b)
)ζμ(τ )
(p �(τ )
p(τ )
)�τ
= log (
p(b)
p(s )
)+ log
(p(t)
p(σ (b))
)+
∫ σ (b) b
ζμ(τ )
(p �(τ )
p(τ )
)�τ
= log (
p(b)
p(s )
)+ log
(p(t)
p(c)
)+ μ(b) ζμ(b)
(p �(b)
p(b)
)= log
(p(b)
p(s )
)+ log
(p(t)
p(c)
)+ μ(b)
[1
μ(b) log
(1 + μ(b) p
�(b)
p(b)
)]= log
(p(b)
p(s )
)+ log
(p(t)
p(c)
)+ log
(p σ (b)
p(b)
)= log
(p(b)
p(s )
)+ log
(p(t)
p(c)
)+ log
(p(c)
p(b)
)= log
(p(t)
p(s )
).
Consequently, in all cases, we see that p (t, s ) = log (
p(t) p(s )
)on this time scale as well.
Example 7.4. Let T = (−∞ , −4] ∪ [2 , ∞ ) , and p(t) = t 3 . Let
t ≥ 2 and s = −5 . Then μ(−4) = σ (−4) − (−4) = 2 − (−4) = 6 ,
and the principal logarithm on this time scale is
L p (t, s ) = L p (t, −5) = ∫ t
−5 ξμ(τ )
((τ 3 ) �
τ 3
)�τ
= (∫ −4
−5 +
∫ 2 −4
+ ∫ t
2
)ξμ(τ )
(σ (τ ) 2 + τσ (τ ) + τ 2
τ 3
)�τ
= 3 (∫ −4
−5 +
∫ t 2
)d τ
τ+ μ(−4) ξμ(−4)
(2 2 − 4(2) + (−4) 2
(−4) 3 )
= 3 ( Log [ −4] − Log [ −5] + Log [ t] − Log [2] ) + Log (
1 + 6 12 −64 )
= 3 ln (
t
5
)+ iπ,
where Log again represents the principal complex logarithm, and
ln is the natural logarithm. Again for sake of comparison,
the logarithms in [10] and [5, Section 3] do not apply as they
are defined exclusively in terms of p(t) = t, and [13] does
notapply as that logarithm requires 1 ∈ T . If we use the logarithm
in [5, Section 4] or [12] , we get 3 ln
(2 t 5
)− 9 8 , a real-valued
function, as opposed to our principal value of 3 ln (
t 5
)+ iπ, a complex-valued function. This example justifies our
approach.
Example 7.5. Here is an example of Theorem 3.3 . Let t ∈ T with
t � = 0 , and set p(t ) = t . For s ∈ T , we have
�p (t, s ) =
⎧ ⎪ ⎨ ⎪ ⎩ 1
μ(t) log
(σ (t)
t
)for μ(t) � = 0
1
t for μ(t) = 0 ,
where �-differentiation is with respect to t . Thus,
�p (t, s ) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1
t for T = R
1
h log
(1 + h
t
)for T = h Z
log (q )
(q − 1) t for T = q N 0 ,
11
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D.R. Anderson and M. Bohner Applied Mathematics and Computation
397 (2021) 125954
Fig. 2. A plot of the function 1 t
versus Log (1 + 1
t
)for Example 7.5 .
where h > 0 and q > 1 .
See Fig. 2 for h = 1 and T = Z . This ends the example. Example
7.6. Construct a discrete time scale with two step sizes that
alternate; that is, for the two alternating step sizes
γ , υ > 0 with γ � = υ, let T := T γ ,υ = { 0 , γ , (γ + υ) ,
(γ + υ) + γ , 2(γ + υ) , 2(γ + υ) + γ , 3(γ + υ) , · · · } .
Then, for t ∈ T and k ∈ N 0 = { 0 , 1 , 2 , . . . } , we
have
μ(t) = {γ for t = k (γ + υ) , υ for t = k (γ + υ) + γ .
Set p(t) = t . We claim that for t ∈ T γ ,υ with t � = 0 ,
�p (t, s ) =
⎧ ⎨ ⎩ 1
γlog
(1 + γ
t
)for t = k (γ + υ)
1
υlog
(1 + υ
t
)for t = k (γ + υ) + γ .
To verify this, note that
�p (t, s ) = 1
μ(t) log
(σ (t)
t
)
=
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1
γlog
(k (γ + υ) + γ
k (γ + υ)
)for t = k (γ + υ)
1
υlog
((k + 1)(γ + υ) k (γ + υ) + γ
)for t = k (γ + υ) + γ
=
⎧ ⎨ ⎩ 1
γlog
(1 + γ
t
)for t = k (γ + υ)
1
υlog
(1 + υ
t
)for t = k (γ + υ) + γ .
This ends the example.
Remark 7.7. The first three examples, given above, suggest that
this new logarithm may be a kind of exact discretization,
in other words, that, by definition it yields the usual
logarithm function restricted to the given time scale. This remains
an
open question for more intricate and general time scales.
Acknowlgedgments
Dedicated to Professor Allan C. Peterson, our mentor, colleague,
and friend, on the occasion of his retirement after 51
years at the University of Nebraska-Lincoln.
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A multivalued logarithm on time scales1 Introduction2 A new
logarithm on time scales3 Properties of the logarithm4 The nabla
case5 Logarithms for Cayley-exponential functions6 Previous
logarithms on time scales7 Numerical comparisons and examples of
logarithmsAcknowlgedgmentsReferences