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Generalized Trigonometric Functions and Elementary Applications

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Abstract

We present some applications of the generalized trigonometric functions to problems in classical mechanics and to the theory of integral equations. We discuss how second and third order trigonometries are ideally suited tools to treat either damped harmonic oscillators and three dimensional rotational models. We make further progress in the generalization process by discussing the properties of Laguerre trigonometries along with the relevant link with the theory of Bessel functions.

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Notes

  1. Whenever necessary to emphasize the dependence on the constants \(a,\, b\) we will use the notation \(C(\theta | a,b),\, S(\theta | a,b)\).

  2. The Trig-functions in Eq. (2) are neither even nor odd under parity transformation and it is understood that \(C(-\theta | a,b)=C(\theta | a,-b)\) and \(S(-\theta | a,b) = -S(\theta | a,-b)\).

  3. It is worth noticing that \(e^{n h \theta }=C(n\theta )+h S(n\theta )=\left[ C(\theta )+h S(\theta )\right] ^{n}\), which can be exploited to derive identities mimicking those of the circular (and hyperbolic) trigonometry. Regarding the duplication formulae we find \(e^{2 h \theta }= C^2(\theta )+a S^2(\theta )+ 2h C(\theta )S(\theta )\), eventually providing the second degree trigonometric duplication formula \(C(2\theta ) = C^2(\theta )+a S^2(\theta )\), \(S(2\theta ) = 2 S(\theta )C(\theta )+b S^2(\theta )\).

  4. In the case of the ordinary imaginary unit \(\phi ^*={\varvec{i}}\pi /2\).

  5. For integer \(\nu \), Eq. (48) is nothing but a Cauchy-repeated integral.

  6. It can be expressed in terms of the \(0-th\) order modified Bessel through the identity \({}_l e(\eta )=I_0(2\sqrt{\eta })\).

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Acknowledgments

We would like to thank Danilo Babusci and Robert Yamaleev for their kind interest and encouragement in this work.

Author information

Authors and Affiliations

  1. ENEA Fusion Department, Via E. Fermi 45, 00044, Frascati, Rome, Italy

    Giuseppe Dattoli, Emanuele Di Palma & Federico Nguyen

  2. ENEA Fusion Department, Piazzale E. Fermi 1, 80055, Portici, Naples, Italy

    Elio Sabia

Authors
  1. Giuseppe Dattoli
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  2. Emanuele Di Palma
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  3. Federico Nguyen
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  4. Elio Sabia
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Corresponding author

Correspondence to Federico Nguyen.

Appendix

Appendix

In this appendix we will clarify the role and the genesis of the composition rule of Eq. (75).

The Laguerre derivative satisfies the identity

$$\begin{aligned} {}_l\partial _\xi ^{n} = \partial _\xi ^{n}\xi ^{n}\partial _\xi ^{n} \end{aligned}$$
(79)

and, accordingly, the pseudo-shift operator \(e^{\hat{c}y {}_l\partial _x}={}_l e(y {}_l \partial _x)\) acting on a monomial \(x^n\) yields

$$\begin{aligned} {}_l e(y {}_l\partial _{x}) x^{n} = \left[ \sum _{r=0}^\infty \frac{y^r}{(r!)^2} \partial _{x}^{r} x^{r}\partial _{x}^{r}\right] x^{n} = \sum _{r=0}^{n}\frac{y^{r}}{(r!)^2}\frac{(n!)^2}{\left[ (n-r)!\right] ^2} x^{n-r} = (x\oplus _l y)^{n} \end{aligned}$$
(80)

It can now be argued that the composition of Eq. (75) can be considered as the Newton binomial associated with the algebraic structure due to the specific form of the operator \(\hat{c}^{n}\).

According to the previous identities we can also state that

$$\begin{aligned} {}_l e(y {}_l\partial _{x}) {}_l e(x)= & {} {}_l e(y) {}_l e(x) \nonumber \\ {}_l e(y {}_l\partial _{x}) {}_l e(x)= & {} {}_l e(x\oplus _l y) \end{aligned}$$
(81)

allowing the derivation of the following semi-group property of the l-exponential

$$\begin{aligned} {}_l e(y) {}_l e(x) = { }_l e(x\oplus _l y) \end{aligned}$$
(82)

In full analogy with the ordinary Euler formula we introduce the l-trigonometric (l-t)-functions through the identity

$$\begin{aligned} {}_l e({\varvec{i}}x) = {}_l c(x)+{\varvec{i}}~ {}_l s(x) \end{aligned}$$
(83)

where the l-t cosine and sine functions are specified by the series

$$\begin{aligned} {}_l c(x)= & {} \sum _{r=0}^\infty \frac{(-1)^r x^{2 r}}{[(2 r)!]^2} \nonumber \\ {}_l s(x)= & {} \sum _{r=0}^\infty \frac{(-1)^r x^{2r+1}}{[(2r+1)!]^2} \end{aligned}$$
(84)

It is easily checked that they satisfy the identities

$$\begin{aligned} {}_l\partial _x\left[ {}_l c(\alpha x)\right]= & {} -\alpha ~ {}_l s(\alpha x)\nonumber \\ {}_l\partial _x\left[ {}_l s(\alpha x)\right]= & {} \alpha ~ {}_l c(\alpha x) \end{aligned}$$
(85)

and therefore the harmonic equation

$$\begin{aligned} ({}_l\partial _x)^2\left[ {}_l c(\alpha x)\right]= & {} -\alpha ^2 {}_l c(\alpha x)\nonumber \\ ({}_l\partial _x)^2\left[ {}_l s(\alpha x)\right]= & {} -\alpha ^2 {}_l s(\alpha x) \end{aligned}$$
(86)

The use of the properties of the l-exponential function allows the derivation of the following addition theorems for the functions in Eq. (84)

$$\begin{aligned} {}_l c(x\oplus _l y)= & {} {}_l c(x) {}_l c(y)-{}_l s(x) {}_l s(y)\nonumber \\ {}_l s(x\oplus _l y)= & {} {}_l c(x) {}_l s(y)+{}_l s(x) {}_l c(y) \end{aligned}$$
(87)

The proof is given below, by observing that

$$\begin{aligned}&{}_l e({\varvec{i}}x) {}_l e({\varvec{i}}y)=\left[ {}_l c(x)+{\varvec{i}}~ {}_l s(x)\right] \left[ {}_l c(y)+{\varvec{i}}~ {}_l s(y)\right] \nonumber \\&\qquad =\left[ {}_l c(x) {}_l c(y)-{}_l s(x) {}_l s(y)\right] +{\varvec{i}}\left[ {}_l c(x) {}_l s(y)+{}_l s(x) {}_l c(y)\right] \end{aligned}$$
(88)

and since

$$\begin{aligned} {}_l e({\varvec{i}}x) {}_l e({\varvec{i}}y) = {}_l e\left( {\varvec{i}}(x\oplus _l y)\right) ={}_l c(x\oplus _l y)+{\varvec{i}}~ {}_l s(x\oplus _l y) \end{aligned}$$
(89)

we can equate real and imaginary parts to infer the identities of Eq. (87).

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Dattoli, G., Palma, E.D., Nguyen, F. et al. Generalized Trigonometric Functions and Elementary Applications. Int. J. Appl. Comput. Math 3, 445–458 (2017). https://doi.org/10.1007/s40819-016-0168-5

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