Abstract
The chapter describes our approach, i.e., the multivariate power substitution and its variations, illustrated by many interesting examples involving commonly used special functions along with applications of the Laplace transform pairs of relevant functions, and the use of permutation symmetry. The combination of the multivariate power substitution and permutation is shown to be a powerful tool to generate interesting integral identities.
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Notes
- 1.
Functions with n th roots are in general more difficult to integrate; classically the power substitution converts these n th roots into integer powers easier to integrate due to the power rule of integration.
- 2.
The error function is important in many fields, e.g., probability theory, heat conduction, and fluid mechanics. It is defined by \(\mathrm {erf}(x):=\frac {2}{\sqrt {\pi }}\int _0^z e^{-t^2}dt\). The complementary error function is given by erfc(z) := 1 −erf(z). J.W.L. Glaisher (1848–1928) coined the term “error function,” reflecting its utility for computing the probability of “errors” from a normal distribution.
- 3.
The development of the gamma function has received great contributions from many eminent mathematicians, to include Adrien-Marie Legendre (1752–1833), Carl Friedrich Gauss (1777–1855), Christopher Gudermann (1798–1852), Joseph Liouville (1809–1882), Karl Weierstrass (1815–1897), Charles Hermite (1822–1901). The notation Γ(x) is due to Legendre in 1809, Gauss preferring \(\prod (x)\), which is actually Γ(x + 1).
- 4.
The confluent hypergeometric function of the second kind U(a, b, c) was introduced by Francesco Tricomi (1897–1978) as a solution to Kummer’s differential equation, i.e., zw″ + (b − z)w′− aw = 0.
- 5.
Named for Friedrich Wilhelm Bessel (1784–1846), Bessel functions were in fact were introduced by Daniel Bernoulli in 1731 and were used by Leonhard Euler to analyze the vibrations of a circular membrane. Andrew Gary and George Matthews developed applications of Bessel functions for electricity, hydrodynamics, and diffraction in 1895, followed by a major treatise on Bessel functions in 1922 by Watson. The Bessel functions of the first and second kind, respectively denoted Jν and Yν, are in fact the two linearly independent solutions of the Bessel differential equation, i.e., x 2 y″ + xy′ + (x 2 − ν 2)y = 0. Setting \(x=i\hat {x}\) leads to the so-called modified Bessel functions of the first and second kind, respectively denoted by Iν and Kν.
- 6.
Giulio Fagnano (1682–1766) did important early work involving transformations of elliptic integrals.
- 7.
The polygamma function is given by \(\psi _n(x)=\frac {d^{n+1}}{dx^{n+1}}\ln \Gamma (x)\).
- 8.
Named after Carl Friedrich Gauss (1777–1855).
- 9.
Named after Cornelis Simon Meijer (1904–1974), the Meijer G-function generalizes most elementary functions and special functions.
- 10.
Here we depart from the common notation, i.e., \(F(s)=\int _0^\infty f(t)e^{-st}dt; t > 0\), which is typically used for the processing of time-dependent signals.
- 11.
The generalized hypergeometric function is an extension of the Gaussian hypergeometric function.
- 12.
Introduced by Leo August Pochhammer (1841–1920).
- 13.
Named after Augustin-Jean Fresnel (1788–1827), the Fresnel integrals are used to study wave propagation.
- 14.
Leonhard Euler introduced γ in 1735, defining it as \(\gamma :=\lim _{n\rightarrow \infty }\left \{\sum _{k=1}^n \frac {1}{k}-\ln n\right \}\).
- 15.
The exponential integral function was introduced by Adrien-Marie Legendre (1752–1833).
- 16.
Named after Gottfried Wilhelm Leibniz (1646–1716).
- 17.
See R. P. Feynman et al., Surely You’re Joking, Mr. Feynman!, W. W. Norton, 1985.
- 18.
Gottfried Wilhelm Leibniz (1646-1716) first defined Li2(z), also known as the dilogarithm.
- 19.
The incomplete beta function is a generalization of the beta function, which is given by \(B(\alpha ,\beta )=\frac {\Gamma (\alpha )\Gamma (\beta )}{\Gamma (\alpha +\beta )}=\int _0^1 u^{\alpha -1}(1-u)^{\beta -1}du\). Among other things, the beta function played an important role in the initial development of string theory.
References
Andrews, G.E. “Applications of Basic Hypergeometric Functions.” SIAM Review 16, (1974), 441–484.
Andrews, G.E., Askey, R. and Roy, R. “Special Functions.” Encyclopedia of Mathematics and its Applications. Cambridge University Press, New York (1999)
Lax, P.D. “Change of variable in multiple integrals II.” Amer. Math. Monthly, 108 (2001), 115–119.
Liu, Shibo and Zhang, Yashan “On the Change of Variables Formula for Multiple Integrals.” J. Math. Study, 50 (2017), 268–276.
Ng, Edward W., and Murray Geller. “A table of integrals of the error functions.” Journal of Research of the National Bureau of Standards B 73, no. 1 (1969): 1–20.
Olver, F., Lozier, D., Boisvert, R., Clark, C. (Eds.) NIST Handbook of Mathematical Functions, U.S. Department of Commence and Cambridge University Press, Cambridge (2010)
Parmar, R.K. “A new generalization of gamma, beta hypergeometric and confluent hypergeometric functions.” Le Matematiche 68, no. 2 (2013): 33–52.
Pham-Gia, T. and Thanh, D.N. “Hypergeometric functions. From one scalar variable to several matrix arguments in statistics and beyond.” Open Journal of Statistics, 6 no 05(2016), 951
Strook, D. “A concise introduction to the Theory of Integration.” 2nd Edition, Birkhauser (1994).
Temme, N.M. “Special Functions. An introduction to the classical Functions of Mathematical Physics.” John Wiley and Sons, New York (1996).
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Ruffa, A.A., Toni, B. (2022). The Multivariate Power Substitution and Its Variants. In: Innovative Integrals and Their Applications I. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-031-17871-9_2
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