1/11 NuttanonSongsuwan(Z118170)SITSupervisor:Prof.ShingoTakeuchiDepartmentofMathematicsKingMongkut’sUniversityofTechnologyThonburi sin (2 x )) Double-angleformulaofgeneralizedtrigonometricfunctionsforsomespecialcase(

Project Abstract Introduction Preliminaries Main Result Conclusion

Double-angle formula of generalized trigonometric functions for some special case (sin

3,4

(2x))

Nuttanon Songsuwan (Z118170) SIT Supervisor: Prof. Shingo Takeuchi

Department of Mathematics

King Mongkut’s University of Technology Thonburi

Project Abstract Introduction Preliminaries Main Result Conclusion

Abstract

In this presentation, we will introduce the double-angle formula for the very special case of the generalized trigonometric functions (sin

) when p =

and q = 4, which is discovered by D.E. Edmunds, P. Gurka, and J. Lang [1]. We also give the alternative proof of the double-angle formula of sin

3,4

, which is introduced by S. Takeuchi [2].

Reference

[1] David E. Edmunds, Petr Gurka, and Jan Lang,Properties of generalized trigonometric functions, Journal of Approximation Theory164(2012), no. 1, 47 –56.

[2] Shingo Takeuchi,Multiple-angle formulas of generalized trigonometric functions with two parameters, Journal of Mathematical Analysis and Applications444(2016), no. 2, 1000 –1014.

Project Abstract Introduction Preliminaries Main Result Conclusion

Generalized trigonometric functions

Generalized trigonometric functions

From the calculus class, we know that sin

x =

Z x

0

(1

t

)

Now we want to do the generalization of the trigonometric function by defined

: [0, 1]

[0,

F_p,q

(x) :=

Z x

0

(1

t

)

−1 p dt,

where p, q

1.Since

is strictly increasing (F

is injective) therefore

is invertible and we denote

sin

:=

Project Abstract Introduction Preliminaries Main Result Conclusion Generalized trigonometric functions

The sin

is defined on an interval [0,

], where

:= 2F

(1) = 2

Z 1 0

(1

t

)

We can extend sin

to the whole real line (

) by using Symmetry and Periodicity property. Now we define cos

:

by

cos

x :=

dx

sin

x.

From this, we get the following property

|cos_p,q

x|

+

x

= 1 for x

Project Abstract Introduction Preliminaries Main Result Conclusion Importance Lemma and Theorem

For the ordinary trigonometric functions, we know the addition formulae (sin(x + y)) and double-angle formulae (sin(2x)) in term of sin(x) and sin(y) as

sin(x + y) = sinxcosy + cosxsiny and sin(2x) = 2sinxcosx

Unfortunately, we do not know the GENERAL addition formula or double-angle

formula for the generalized trigonometric functions.

Project Abstract Introduction Preliminaries Main Result Conclusion Importance Lemma and Theorem

We define p

:=

. For p = 2 and q = 4, the generalized trigonometric function sin

x coincides with the lemniscate (sine) function

Lemma (The double-angle formula of the lemniscate function) Let

be the lemniscate function and x

[0,

], then

sl(2x) =

2slx

1

x

1 +

x ;

sin

x. (DL)

Theorem (The multiple-angle formula of generalized trigonometric funtion [2]) For p

1 and x

[0,

] = [0,

], we have

sin

(2

x) = 2

sin

xcos

x, (MG)

Project Abstract Introduction Preliminaries Main Result Conclusion The double formula for thesin_4/3,4

In 2011, D.E. Edmunds, P. Gurka, and J. Lang [1] discovered the double-angle formula for some very special case of the generalized trigonometric function (sin

) when p = 4 .

Theorem (The double-angle formula for

3,4x

[1]) Let x

[0,

]. Then

sin

3,4

(2x) = 2sin

x(cos

x) 1 + 4(sin

x)(cos

x)

Their proof of this theorem relies on the Jacobian elliptic functions. So we will

introduce the alternative proof by using the lemniscate (sine) function. This proof was

discovered by S. Takeuchi [2].

Project Abstract Introduction Preliminaries Main Result Conclusion The double formula for thesin_4/3,4

Sketch of Proof.

Applying (MG) in case p = 4 with x replaced by 2x

[0,

), we get

√

2x) =

√

2sin

(2x)(1

sin

(2x))

This equation gives

2sin

(2x ) = 1

1

(2

√

2x).

Set S = S(x) :=

2x). Using (DL), we have

2sin

(2x) = 1

t

1

2S

1

S

1 + S

!4

= 1

6S

+ S

(1 + S

)

Project Abstract Introduction Preliminaries Main Result Conclusion The double formula for thesin_4/3,4

Sketch of Proof (Cont.)

We see that when 0

x

, 1

6S

+ S

0 and when

x

, 1

6S

+ S

0. Thus,

2sin

(2x ) = 1

1

6S

+ S

(1 + S

)

= 8S

(1 + S

)

Therefore, by (MG),

sin

(2x) =

√

2S

√

1 + S

= 2sin

x(cos

x)

1 + 4(sin

x)(cos

x)

Project Abstract Introduction Preliminaries Main Result Conclusion

Conclusion

By studying the double-angle formula for the special case of generalized trigonometric function (sin

) when p = 4, which was discovered by D.E. Edmunds, P. Gurka, and J. Lang [1] in 2011, and the proof by using lemniscate (sine) function, which was introduced by S. Takeuchi [2], in 2016, I have learned several related topics such as

The properties of the generalized trigonometric functions, Jacobian elliptic functions, and

The properties of beta and gamma functions.

Project Abstract Introduction Preliminaries Main Result Conclusion

Acknowledgement

I would like to express my thanks to : King Mongkut’s University of Technology

Professor Shingo Takeuchi

Shibaura Institute of Technology

All of my colleagues in Functional

Equations Laboratory