applied mathematics and computationd.r. anderson and m. bohner applied mathematics and computation...

Applied Mathematics and Computation 397 (2021) 125954

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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 ,

2


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 ,

3


(

(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


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] .

5


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 ,

6


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

7


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.

8


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


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 )

).

10


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


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.

References

[1] D.R. Anderson , S.G. Georgiev , Conformable Dynamic Equations on Time Scales, Chapman and Hall/CRC, Boca Raton, 2020 . [2] M. Bohner , S.G. Georgiev , Multivariable Dynamic Calculus on Time Scales, Springer International Publishing, 2016 .

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[11] B. Ito, P. R ̆ehák, N. Yamaoka, Applications of iterated logarithm functions on time scales to Riemann-weber-type equations, 2020. Proc. Amer. Math.Soc., 148, 1611–1624, 10.1090/proc/14812.

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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

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