Skip to main content
SpringerLink
Log in

Addition theorems for \({\mathcal {C}}^k\) real functions and applications in ordinary differential equations

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

This work establishes the existence of addition theorems and double-angle formulas for \({\mathcal {C}}^k\) real scalar functions. Moreover, we determine necessary and sufficient conditions for a bivariate function to be an addition formula for a \({\mathcal {C}}^k\) real function. The double-angle formulas allow us to generate a duplication algorithm, which can be used as an alternative to the classical numerical methods to obtain an approximation for the solution of an ordinary differential equation. We demonstrate that this algorithm converges uniformly in any compact domain contained in the maximal domain of that solution. Finally, we carry out some numerical simulations showing a good performance of the duplication algorithm when compared with standard numerical methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Aczél, J.: Lectures on Functional Equations and Their Applications, Mathematics in Science and Engineering, vol. 19. Academic Press, New York (1966)

    MATH  Google Scholar 

  2. Bukhshtaber, V.M., Krichever, I.M.: Vector addition theorems and Baker–Akhiezer functions. Theor. Math. Phys. 94, 142–149 (1993)

    Article  MathSciNet  Google Scholar 

  3. Bukhshtaber, V.M., Krichever, I.M.: Multidimensional vector addition theorems and the Riemann theta functions. Int. Math. Res. Not. 10, 505–513 (1996)

    Article  MathSciNet  Google Scholar 

  4. Bulirsch, R.: Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7(1), 78–90 (1965)

    Article  MathSciNet  Google Scholar 

  5. Bulirsch, R.: Numerical calculation of elliptic integrals and elliptic functions III. Numer. Math. 13(4), 305–315 (1969)

    Article  MathSciNet  Google Scholar 

  6. Carlson, B.C.: Computing elliptic integrals by duplication. Numer. Math. 33(1), 1–16 (1979)

    Article  MathSciNet  Google Scholar 

  7. Fukushima, T.: Precise and fast computation of elliptic integrals and elliptic functions. In: IEEE 22nd Symposium on Computer Arithmetic (2015)

  8. Hancock, H.: Lectures on the Theory of Elliptic Functions. Wiley, Hoboken (1910)

    MATH  Google Scholar 

  9. Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions. Birkhäuser Advanced Texts (2002)

  10. Kuwagaki, A.: Sur la fonction analytique de deux variables complexes satisfaisant l’associativité: \(f\{x, f(y, z)\}=f\{f(x, y), z\}\). Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 27(3), 225–234 (1953)

    MathSciNet  MATH  Google Scholar 

  11. Lawden, D.F.: Elliptic Functions and Applications, Applied Mathematical Sciences, vol. 80. Springer, Berlin (1989)

    Book  Google Scholar 

  12. Montel, P.: Sur les fonctions d’une variable réelle qui admettent un théorème d’addition algébrique. Annales scientifiques de l’École Normale Supérieure, 3e série 48, 65–94 (1931)

    Article  Google Scholar 

  13. Painleve, P.: Sur les fonctions qui admettent un théoréme d’addition. Acta Math. 26, 1–54 (1902)

    Article  MathSciNet  Google Scholar 

  14. Phragmen, E.: Sur un théoréme concernant les fonctions elliptiques. Acta Math. 7, 33–42 (1885)

    Article  MathSciNet  Google Scholar 

  15. Ritt, J.F.: Real functions with algebraic addition theorems. Trans. Am. Math. Soc. 29(2), 361–368 (1927)

    Article  MathSciNet  Google Scholar 

  16. Tsiganov, A.V.: Leonard Euler: addition theorems and superintegrable systems. Regul. Chaotic Dyn. 14(3), 389–406 (2009)

    Article  MathSciNet  Google Scholar 

  17. Ungar, A.: Addition theorems for solutions to linear homogeneous constant coefficient ordinary differential equations. Aequ. Math. 26, 104–112 (1983)

    Article  MathSciNet  Google Scholar 

  18. Ungar, A.: Addition theorems in ordinary differential equations. Am. Math. Mon. 94(9), 872–875 (1987)

    Article  MathSciNet  Google Scholar 

  19. Weierstrass, K.: Formeln uns Lehrstze sum Gebrauche des elliptischen Functionen (1855)

  20. Zapata, J.: Analytic and geometric techniques for non-integrable dynamics systems. Applications to perturbed Keplerian models. Ph.D. thesis, Universidad del Bío-Bío (2020)

Download references

Acknowledgements

Support from Research Agencies of Chile is acknowledged. They came in the form of research Projects 11160224 of the Chilean national agency FONDECYT and the UBB Project 2020157 IF/R. The author J.L.Z. acknowledges support from CONICYT PhD/2017-21170836 and Proyecto Plurianual AIUE 1955 UBB.

Author information

Authors and Affiliations

  1. GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Av. Collao 1202, Casilla 5-C, Concepción, Chile

    Francisco Crespo, Salomón Rebollo-Perdomo & Jorge L. Zapata

Authors
  1. Francisco Crespo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Salomón Rebollo-Perdomo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jorge L. Zapata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jorge L. Zapata.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Double-angle formula Taylor expansion

Appendix A: Double-angle formula Taylor expansion

The Taylor expansion of the double-angle formula is obtained by following the strategy given in Sect. 3. Here we include the expression of the derivatives up to the 10th order

$$\begin{aligned} R(x_0)= & {} x_0,\ \ R^{(1)}(x_0)= 2,\ \ R^{(2)}(x_0)=\frac{2f^{(1)}(x_0)}{f(x_0)},\ \ R^{(3)}(x_0)= \frac{ 6 f^{(2)}(x_0)}{f(x_0)},\\ R^{(4)}(x_0)= & {} \frac{2}{f(x_0)^2}\left( 7 f^{(3)}(x_0) f(x_0)+6 f^{(1)}(x_0) f^{(2)}(x_0) \right) ,\\ R^{(5)}(x_0)= & {} \frac{30 \left( f(x_0) f^{(4)}(x_0)+2 f^{(2)}(x_0)^2+2 f^{(3)}(x_0) f^{(1)}(x_0)\right) }{f(x_0)^2},\\ R^{(6)}(x_0)= & {} \frac{1}{f(x_0)^3} \left( 62 f^{(5)}(x_0) f(x_0)^2+20 \left( 28 f(x_0) f^{(3)}(x_0) f^{(2)}(x_0)\right. \right. \\&\left. \left. +3 f^{(3)}(x_0) f^{(1)}(x_0)^2+f^{(1)}(x_0) \left( 11 f(x_0) f^{(4)}(x_0)+9 f^{(2)}(x_0)^2\right) \right) \right) ,\\ R^{(7)}(x_0)= & {} \frac{14}{f(x_0)^3} \left( 9 f^{(6)}(x_0) f(x_0)^2+10 \left( 9 f^{(2)}(x_0)^3+4 f^{(4)}(x_0) f^{(1)}(x_0)^2\right. \right. \\&\left. \left. +5 f(x_0) \left( 2 f^{(3)}(x_0)^2+f^{(5)}(x_0) f^{(1)}(x_0)\right) \right. \right. \\&\left. \left. +\left( 15 f(x_0) f^{(4)}(x_0)+23 f^{(3)}(x_0) f^{(1)}(x_0)\right) f^{(2)}(x_0)\right) \right) ,\\ R^{(8)}(x_0)= & {} \frac{1}{f(x_0)^4}\left( 254 f^{(7)}(x_0) f(x_0)^3+28 \left( 20 f^{(4)}(x_0) f^{(1)}(x_0)^3\right. \right. \\&\left. \left. +f(x_0) \left( 251 f(x_0) f^{(5)}(x_0) f^{(2)}(x_0)+5 f^{(3)}(x_0)\right. \right. \right. \\&\left. \left. \left. \quad \left( 81 f(x_0) f^{(4)}(x_0)+199 f^{(2)}(x_0)^2\right) \right) \right. \right. \\&\left. \left. +5 f^{(1)}(x_0)^2 \left( 23 f(x_0) f^{(5)}(x_0)+32 f^{(3)}(x_0) f^{(2)}(x_0)\right) +f^{(1)}(x_0) \right. \right. \\&\quad \left. \left. \left( 180 f^{(2)}(x_0)^3\right. \right. \right. \\&\left. \left. \left. +f(x_0) \left( 73 f(x_0) f^{(6)}(x_0)+455 f^{(3)}(x_0)^2\right) \right. \right. \right. \\&\left. \left. \left. +715 f(x_0) f^{(4)}(x_0) f^{(2)}(x_0)\right) \right) \right) ,\\ R^{(9)}(x_0)= & {} \frac{1}{f(x_0)^4}\!\left( \!510 f^{(8)}(x_0) f(x_0)^3\!+\!84 \left( \!540 f^{(2)}(x_0)^4+90 f^{(5)}(x_0) f^{(1)}(x_0)^3 \right. \right. \\&\left. \left. +6 \left( 29 f(x_0) f^{(6)}(x_0)+75 f^{(3)}(x_0)^2\right) f^{(1)}(x_0)^2 \right. \right. \\&\left. \left. +f(x_0)^2 \left( 295 f^{(4)}(x_0)^2+488 f^{(3)}(x_0) f^{(5)}(x_0)\right) \right. \right. \\&\left. \left. +f(x_0) \left( 67 f(x_0) f^{(7)}(x_0)+1820 f^{(3)}(x_0) f^{(4)}(x_0)\right) f^{(1)}(x_0) \right. \right. \\&\left. \left. +270 \left( 7 f(x_0) f^{(4)}(x_0)+9 f^{(3)}(x_0) f^{(1)}(x_0)\right) f^{(2)}(x_0)^2 \right. \right. \\&\left. \left. +2 \left( 405 f^{(4)}(x_0) f^{(1)}(x_0)^2+f(x_0) \left( 130 f(x_0) f^{(6)}(x_0) \right. \right. \right. \right. \\&\left. \left. \left. \left. +1235 f^{(3)}(x_0)^2+603 f^{(5)}(x_0) f^{(1)}(x_0)\right) \right) f^{(2)}(x_0)\right) \right) ,\\ R^{(10)}(x_0)= & {} \frac{2}{f(x_0)^5} \left( 511 f^{(9)}(x_0) f(x_0)^4+3780 f^{(5)}(x_0) f^{(1)}(x_0)^4 \right. \\&\left. +252 f^{(1)}(x_0)^3 \left( 117 f(x_0) f^{(6)}(x_0)+75 f^{(3)}(x_0)^2+175 f^{(4)}(x_0) f^{(2)}(x_0)\right) \right. \\&\left. +12 f(x_0) \left( 78330 f^{(3)}(x_0) f^{(2)}(x_0)^3+7 f(x_0) \left( 3170 f^{(3)}(x_0)^3 \right. \right. \right. \\&\left. \left. \left. +4647 f^{(5)}(x_0) f^{(2)}(x_0)^2+14440 f^{(4)}(x_0) f^{(3)}(x_0) f^{(2)}(x_0)\right) \right. \right. \\&\left. \left. +f(x_0)^2 \left( 2679 f^{(7)}(x_0) f^{(2)}(x_0)+5726 f^{(3)}(x_0) f^{(6)}(x_0) \right. \right. \right. \\&\left. \left. \left. +8029 f^{(4)}(x_0) f^{(5)}(x_0)\right) \right) +42 f^{(1)}(x_0)^2 \left( 7158 f(x_0) f^{(5)}(x_0) f^{(2)}(x_0) \right. \right. \\&+4350 f^{(3)}(x_0) f^{(2)}(x_0)^2+f(x_0) \Big (683 f(x_0) f^{(7)}(x_0)\\&+9530 f^{(3)}(x_0) f^{(4)}(x_0)\Big )\Big ) +6 f^{(1)}(x_0) \Big (18900 f^{(2)}(x_0)^4\\&+158340 f(x_0) f^{(4)}(x_0) f^{(2)}(x_0)^2 +14 f(x_0) \Big (2648 f(x_0) f^{(6)}(x_0)\\&+13805 f^{(3)}(x_0)^2\Big ) f^{(2)}(x_0) \\&+f(x_0)^2 \Big (1237 f(x_0) f^{(8)}(x_0)+38115 f^{(4)}(x_0)^2\\&+64974 f^{(3)}(x_0) f^{(5)}(x_0)\Big )\Big )\Big ). \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Crespo, F., Rebollo-Perdomo, S. & Zapata, J.L. Addition theorems for \({\mathcal {C}}^k\) real functions and applications in ordinary differential equations. Aequat. Math. 96, 431–452 (2022). https://doi.org/10.1007/s00010-021-00822-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-021-00822-w

Keywords

Mathematics Subject Classification

Search

Navigation