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Certain integral involving inverse hyperbolic function.

[section]1. Introduction and preliminaries

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler. Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form

f(x) = [[integral].sup.x.sub.c] R(t, [square root of (P(t))]) dt, (1.1)

where R is a rational function of its two arguments, P is a polynomial of degree 3 or 4 with no repeated roots, and c is a constant.

In general, elliptic integrals cannot be expressed in terms of elementary functions. Exceptions to this general rule are when P has repeated roots, or when R(x, y) contains no odd powers of y. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms (i.e. the elliptic integrals of the first, second and third kind).

Besides the Legendre form, the elliptic integrals may also be expressed in Carlson symmetric form. Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz-Christoffel mapping. Historically, elliptic functions were discovered as inverse functions of elliptic integrals.

Incomplete elliptic integrals are functions of two arguments, complete elliptic integrals are functions of a single argument.

Definition 1.1. The incomplete elliptic integral of the first kind F is defined as

F([psi], k) = F([psi]|[k.sup.2]) = F(sin[psi]; k) = [[integral].sup.[psi].sub.0] d[theta]/[square root of (1 - [k.sup.2][sin.sup.2][theta])]. (1.2)

This is the trigonometric form of the integral, substituting t = sin[theta], x = sin[psi], one obtains Jacobi's form:

F(x; k) = [[integral].sup.x.sub.0] dt/[square root of ((1 - [t.sup.2])(1 - [k.sup.2][t.sup.2]))]. (1.3)

Equivalently, in terms of the amplitude and modular angle one has:

F([psi]\[alpha]) = F([psi], sin[alpha]) = [[integral].sup.[psi].sub.0] d[theta]/[square root of (1 - [(sin[theta]sin[alpha]).sup.2])]. (1.4)

In this notation, the use of a vertical bar as delimiter indicates that the argument following it is the "parameter" (as defined above), while the backslash indicates that it is the modular angle. The use of a semicolon implies that the argument preceding it is the sine of the amplitude:

F([psi], sin[alpha]) = F([psi]|[sin.sup.2][alpha]) = F([psi]\[alpha]) = F(sin[psi]; sin[alpha]). (1.5)

Definition 1.2. Incomplete elliptic integral of the second kind E is defined as

E([psi], k) = E([psi]|[k.sup.2]) = E(sin[psi]; k) = [square root of (1 - [k.sup.2][sin.sup.2][theta])] d[theta]. (1.6)

Substituting t = sin[theta] and x = sin[psi], one obtains Jacobi's form:

E(x; k) = [[integral].sup.x.sub.0] [square root of (1 - [k.sup.2][t.sup.2])]/[square root of (1 - [t.sup.2])] dt. (1.7)

Equivalently, in terms of the amplitude and modular angle:

E([psi]\[alpha]) = E([psi], sina) = [[integral].sup.[psi].sub.0] [square root of (1 - [(sin[theta]sin[alpha]).sup.2])] d[theta]. (1.8)

Definition 1.3. Incomplete elliptic integral of the third kind [PI] is defined as

[PI](n; [psi]\[alpha]) = [[integral].sup.[psi].sub.0] 1/[1 - n[sin.sup.2][theta]] d[theta]/[1 - [(sin[theta]sin[alpha]).sup.2]], (1.9)

or

[PI](n; [psi]|m) = [[integral].sup.sin[psi].sub.0] 1/[1 - n[t.sup.2]] dt/(1 - m[t.sup.2])(1 - [t.sup.2]). (1.10)

The number n is called the characteristic and can take on any value, independently of the other arguments.

Definition 1.4. Elliptic Integrals are said to be complete when the amplitude [psi] = [pi]/2 and therefore x = 1.

The complete elliptic integral of the first kind K may thus be defined as

K(k) = [[integral].sup.[pi]/2.sub.0] d[theta]/[square root of (1 - [k.sup.2][sin.sup.2][theta])] = [[integral].sup.1.sub.0]dt/[square root of ((1 - [t.sup.2])(1 - [k.sup.2][t.sup.2]))], (1.11)

or more compactly in terms of the incomplete integral of the first kind as

K(k) = F([pi]/2, k) = F(1; k). (1.12)

It can be expressed as a power series

K(k) = [pi]/2 [[infinity].summation over (n = 0)] [[(2n)!/[2.sup.2n][(n!).sup.2]].sup.2][k.sup.2n] = [pi]/2 [[infinity].summation over (n = 0)][[[P.sub.2n](0)].sup.2][k.sup.2n], (1.13)

where [P.sub.n] is the Legendre polynomial, which is equivalent to

K(k) = [pi]/2 [1 + [(1/2).sup.2][k.sup.2] + [(1.3/2.4).sup.2][k.sup.4] +... + [{(2n - 1)!!/(2n)!!}.sup.2][k.sup.2n] +...], (1.14)

where n!! denotes the double factorial.

In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as

K(k) = [pi]/2 [sub.2][F.sub.1] (1/2, 1/2; 1; [k.sup.2]). (1.15)

The complete elliptic integral of the first kind is sometimes called the quarter period. It can most efficiently be computed in terms of the arithmetic-geometric mean:

K(k) = [[pi]/2]/agm(1 - k,1 + k). (1.16)

The complete elliptic integral of the second kind E is proportional to the circumference of the ellipse C:

C = 4aE(e),

where a is the semi-major axis, e is the eccentricity, and E may be defined as

E(k) = [[integral].sup.[pi]/2.sub.0] [square root of (1 - [k.sup.2][sin.sup.2][theta])] d[theta] = [[integral].sup.1.sub.0] [square root of (1 - [k.sup.2][t.sup.2])]/[square root of (1 - [t.sup.2])] dt, (1.17)

or more compactly in terms of the incomplete integral of the second kind as

E(k) = E([pi]/2, k) = E(1; k). (1.18)

It can be expressed as a power series

E(k) = [pi]/2 [[infinity].summation over (n = 0)] [[(2n)!/[2.sup.2n][(n!).sup.2]].sup.2] [k.sup.2n]/[1 - 2n], (1.19)

which is equivalent to

E(k) = [pi]/2 [1 - [(1/2).sup.2] [k.sup.2]/1 - [(1.3/2.4).sup.2] [k.sup.4]/3 -...- [{(2n - 1)!!/(2n)!!}.sup.2] [k.sup.2n]/[2n - 1] -...]. (1.20)

In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as

E(k) = [pi]/2 [sub.2][F.sub.1] (1/2, - 1/2; 1; [k.sup.2]). (1.21)

The complete elliptic integral of the third kind [PI] can be defined as

[PI](n, k) = [[integral].sup.[pi]/2.sub.0] d[theta]/(1 - n[sin.sup.2][theta])[square root of (1 - [k.sup.2][sin.sup.2][theta])]. (1.22)

Definition 1.5. A generalized hypergeometric function [sub.p][F.sub.q] ([a.sub.1],...[a.sub.p]; [b.sub.1],... bq; z) is a function which can be defined in the form of a hypergeometric series, i.e., a series for which the ratio of successive terms can be written

[c.sub.k + 1]/[c.sub.k] = P(k)/Q(k) = (k + [a.sub.1])(k + [a.sub.2])...(k + [a.sub.p])/(k + [b.sub.1])(K + [b.sub.2])...(k + [b.sub.q])(k + 1)z, (1.23)

where k + 1 in the denominator is present for historical reasons of notation, and the resulting generalized hypergeometric function is written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.24)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.25)

where the parameters [b.sub.1], [b.sub.2], ..., [b.sub.q] are neither zero nor negative integers and p, q are nonnegative integers.

The [sub.p][F.sub.q] series converges for all finite z if p [less than or equal to] q, converges for [absolute value of z] < 1 if p [not equal to] q + 1, diverges for all z, z [not equal to] 0 if p > q + 1.

The [sub.p][F.sub.q] series absolutely converges for [absolute value of z] = 1 if R([zeta]) < 0, conditionally converges for [absolute value of z] = 1, z [not equal to] 0 if 0 [less than or equal to] R([zeta]) < 1, diverges for [absolute value of z] = 1, if 1 [less than or equal to] R([zeta]), [zeta] = [p.summation over (i = 1)][a.sub.i] - [q.summation over (i = 0)][b.sub.i].

The function [sub.2][F.sub.1](a, b; c; z) corresponding to p = 2, q = 1, is the first hypergeometric function to be studied (and, in general, arises the most frequently in physical problems), and so is frequently known as "the" hypergeometric equation or, more explicitly, Gauss's hypergeometric function (Gauss 1812, Barnes 1908). To confuse matters even more, the term "hypergeometric function" is less commonly used to mean closed form, and "hypergeometric series" is sometimes used to mean hypergeometric function.

The hypergeometric functions are solutions of Gaussian hypergeometric linear differential equation of second order

z(1 - z)y" + [c - (a + b + 1)z]y' - aby = 0. (1.26)

The solution of this equation is

y = [A.sub.0][1 + ab/1!c z + a(a + 1)b(b + 1)/2!c(c + 1) [z.sup.2] +...]. (1.27)

This is the so-called regular solution, denoted

[sub.2][F.sub.1](a, b; c; z) = [1 + ab/1!c z + a(a + 1)b(b + 1)/2!c(c + 1) [z.sup.2] +...] = [[infinity].summation over (k = 0)][(a).sub.k][(b).sub.k][z.sup.k]/[(c).sub.k]k!, (1.28)

which converges if c is not a negative integer for all of [absolute value of z] < 1 and on the unit circle [absolute value of z] = 1 if R(c - a - b) > 0.

It is known as Gauss hypergeometric function in terms of Pochhammer symbol [(a).sub.k] or generalized factorial function.

[section]2. Main integrals

[integral] d[phi]/[square root of (1 + x[sinh.sup.-1][phi])] = 1/2 [[(-[sinh.sup.-1][phi]).sup.-m][sinh.sup. -1][[phi].sup.m][GAMMA](m + 1, -[sinh.sup.-1][phi]) -[GAMMA](m + 1, [sinh.sup.-1][phi])] [sub.1][F.sub.0] (1/2; -; -x) + Constant. (2.1)

[integral] d[phi]/[square root of (1 + x[cosh.sup.-1][phi])] = 1/2 [[(-[cosh.sup.-1][phi]).sup.-m][cosh.sup. -1][[phi].sup.m][GAMMA](m + 1, -[cosh.sup.-1][phi]) + [GAMMA](m + 1, [cosh.sup.-1][phi])] [sub.1][F.sub.0] (1/2; -; -x) + Constant. (2.2)

[integral] d[phi]/[square root of (1 + x[sin.sup.-1][phi])] = 1/2 [iota][sin.sup.-1][[phi].sup.m][{[sin.sup. -1][[phi].sup.2]}.sup.-m][[(-[iota][sin.sup.-1][phi]).sup.m][GAMMA](m + 1, [iota][sin.sup.-1][phi]) - [([iota][sin.sup. -1][phi]).sup.m][GAMMA](m + 1, -[iota][sin.sup.-1][phi])] [sub.1][F.sub.0] (1/2; -; -x) + Constant. (2.3)

[integral] d[phi]/[square root of (1 + x[cos.sup.-1][phi])] = 1/2 [cos.sup.-1][[phi].sup.m][{[cos.sup. -1][[phi].sup.2]}.sup.-m][[(-[iota][cos.sup.-1][phi]).sup.m][GAMMA](m + 1, [iota][cos.sup.-1][phi]) + + [([iota][cos.sup.-1][phi]).sup.m][GAMMA](m + 1, -[iota][cos.sup.-1][phi])] [sub.1][F.sub.0] (1/2; -; -x) + Constant. (2.4)

Derivation of (2.1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proceeding the same way one can established the other integrals.

References

[1] Abramowitz Milton and Stegun A. Irene, Handbook of mathematical functions with formulas, graphs, and mathematical tables, New York, Dover, 1965.

[2] Lars Ahlfors, Complex analysis: An introduction to the theory of analytic functions of one complex variable(3rd edition), New York, London, McGraw-Hill, 1979.

[3] Alfred George Greenhill, The applications of elliptic functions, New York, Macmillan, 1892.

[4] L. C. Andrews, Special function of mathematics for engineers, Second Edition, McGrawHill Co Inc., New York, 1992.

[5] Bells Richard Wong Roderick, Special functions, A Graduate Text, Cambridge studies in advanced mathematics, 2010.

[6] Borowski Ephraim and Borwein Jonathan, Mathematics, Collins dictionary(2nd edition), Glasgow: HarperCollins, 2002.

[7] Crandall Richard and Pomeramce Carl, Prime Numbers: A computational perspective, New York, Springer, 2001.

[8] J. Derbyshire, Prime Obsession: Bernhard Riemann and the greatest unsolved problem in Mathematics, New York, Penguin, 2004.

[9] Graham L. Ronald, Knuth E. Donald, Patashnik and Oren, Concrete Mathematics, Reading MA, Addison- Wesley, 1994.

[10] Harris Hancock, Lectures on the theory of elliptic functions, New York, J. Wiley & sons, 1910.

[11] S. G. Krantz, Handbook of complex variables, Boston, MA, Birkhser, 1999.

[12] Lemmermeyer Franz, Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, 2000.

[13] Louis V. King, On the direct numerical calculation Of elliptic functions and integrals, Cambridge University Press, 1924.

[14] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: The art of scientific computing(3rd edition), New York, Cambridge University Press, 2007.

[15] Ramanujan Srinivasa, Collected Papers, Providence RI: AMS /Chelsea, 2000.

[16] Ribenboim, Paulo, The New Book of Prime Number Records, New York, Springer, 1996.

[17] Salahuddin, Hypergeometric form of certain indefinite integrals, Global Journal of Science Frontier Research, 12(2012), 33-37.

[18] R. A. Silverman, Introductory complex analysis, New York, Dover, 1984.

[19] Titchmarsh Edward Charles and Heath-Brown David Rodney, The theory of the Riemann Zeta function(2nd edition), Oxford, 1986.

Salahuddin

Mewar University, Gangrar, Chittorgarh(Rajasthan), India

E-mails: vsludn@gmail.com, sludn@yahoo.com
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Date:Sep 1, 2013
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