On the integration of incomplete elliptic integrals

1994 ◽  
Vol 444 (1922) ◽  
pp. 525-532 ◽  
Keyword(s):  
Closed Form ◽  
Hypergeometric Functions ◽  
Elliptic Integrals ◽  
Fractional Integrals ◽  
Special Cases ◽  
Complete Elliptic Integrals ◽  
Incomplete Elliptic Integrals ◽  
Form Evaluation

We give a closed-form evaluation of Erdélyi-Kober fractional integrals, involving incomplete elliptic integrals of the first kind, F ( φ, k ), and of the second kind, E ( φ, k ), which are integrated either with respect to the modulus or the amplitude. This is made possible by representing F ( φ, k ) and E ( φ, k ) in terms of the Kampé de Fériet double hypergeometric functions. Reduction formulae for these enable us to simplify the solutions for thirteen special cases, including integrals involving complete elliptic integrals. The hypergeometric character of the incomplete integrals is useful for evaluations of other classes of integrals involving F ( φ, k ) and E ( φ, k ).

scholarly journals On a family of logarithmic and exponential integrals occurring in probability and reliability theory

1994 ◽  
Vol 35 (4) ◽  
pp. 469-478 ◽  
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Reliability Theory ◽  
Integral Function ◽  
Integral Representations ◽  
Exponential Integral ◽  
Error Functions ◽  
Special Cases ◽  
Image Position ◽  
Form Evaluation

AbstractWe define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

Extended Mittag-Leffler function and associated fractional calculus operators

2020 ◽  
Vol 27 (2) ◽  
pp. 199-209 ◽  
Author(s):  
Junesang Choi ◽  
Rakesh K. Parmar ◽  
Purnima Chopra
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Fractional Integrals ◽  
Generalized Hypergeometric Function ◽  
Pochhammer Symbol ◽  
Special Cases ◽  
Fractional Calculus Operators ◽  
Fractional Integrals And Derivatives ◽  
Appl Math

AbstractMotivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

The Rectifications of the Cassinian Ovals: Third Round

2020 ◽  
Vol 57 ◽  
pp. 87-98
Author(s):  
Ivaïlo M. Mladenov ◽  
Keyword(s):  
Hypergeometric Functions ◽  
Elliptic Integrals ◽  
Explicit Formulas ◽  
Complete Elliptic Integrals ◽  
Equivalent Expressions

Here we derive a bunch of explicit formulas for the circumferences of all types of Cassinian ovals in terms of the complete elliptic integrals of the first kind and their equivalent expressions in terms of the hypergeometric functions.

scholarly journals Direct Bijective Computation of the Generating Series for 2 and 3-Connection Coefficients of the Symmetric Group

10.37236/3226 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Alejandro H. Morales ◽  
Ekaterina A. Vassilieva
Keyword(s):  
Symmetric Group ◽  
Closed Form ◽  
Main Ingredient ◽  
Irreducible Characters ◽  
Long Cycle ◽  
Connection Coefficients ◽  
Special Cases ◽  
Generating Series ◽  
Orientable Surfaces ◽  
Form Evaluation

We evaluate combinatorially certain connection coefficients of the symmetric group that count the number of factorizations of a long cycle as a product of three permutations. Such factorizations admit an important topological interpretation in terms of unicellular constellations on orientable surfaces. Algebraic computation of these coefficients was first done by Jackson using irreducible characters of the symmetric group. However, bijective computations of these coefficients are so far limited to very special cases. Thanks to a new bijection that refines the work of Schaeffer and Vassilieva, we give an explicit closed form evaluation of the generating series for these coefficients. The main ingredient in the bijection is a modified oriented tricolored tree tractable to enumerate. Finally, reducing this bijection to factorizations of a long cycle into two permutations, we get the analogue formula for the corresponding generating series.

scholarly journals Closed-form evaluation of two-dimensional static lattice sums

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160510 ◽  
Author(s):  
S. Yakubovich ◽  
P. Drygas ◽  
V. Mityushev
Keyword(s):  
Closed Form ◽  
Two Dimensional ◽  
Third Order ◽  
Lattice Sums ◽  
The Third ◽  
Analytical Formulae ◽  
Complete Elliptic Integrals ◽  
Exact Relations ◽  
Form Evaluation ◽  
Static Lattice

Closed-form formulae for the conditionally convergent two-dimensional (2D) static lattice sums S 2 (for conductivity) and T 2 (for elasticity) are deduced in terms of the complete elliptic integrals of the first and second kind. The obtained formulae yield asymptotic analytical formulae for the effective tensors of 2D composites with circular inclusions up to the third order in concentration. Exact relations between S 2 and T 2 for different lattices are established. In particular, the value S 2 = π for the square and hexagonal arrays is discussed and T 2 = π /2 for the hexagonal is deduced.

scholarly journals On the Complete Elliptic Integrals and Babylonian Identity II: An Approximation for the Complete Elliptic Integral of the First Kind

2013 ◽  
Vol 6 ◽  
pp. 47-54
Author(s):  
Edigles Guedes ◽  
Raja Rama Gandhi
Keyword(s):  
Closed Form ◽  
Bessel Functions ◽  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Complete Elliptic Integral ◽  
Complete Elliptic Integrals

Using the Theorem 4 of previous paper, I evaluate the complete elliptic integral of the first kind in approximate analytical closed form, by means of Bessel functions.

scholarly journals String solutions in AdS3 × S3 × S3 × S1 with B-field

2017 ◽  
Vol 32 (01) ◽  
pp. 1750007
Author(s):  
Plamen Bozhilov
Keyword(s):  
Hypergeometric Functions ◽  
Dimensional Space ◽  
Space Time ◽  
Elliptic Integrals ◽  
Conserved Charges ◽  
Text Field ◽  
Angular Momenta ◽  
Dimensional Space Time ◽  
Incomplete Elliptic Integrals

We consider strings living in [Formula: see text] with nonzero [Formula: see text]-field. By using specific ansatz for the string embedding, we obtain a class of solutions corresponding to strings moving in the whole ten-dimensional space–time. For the [Formula: see text] subspace, these solutions are given in terms of incomplete elliptic integrals. For the two three-spheres, they are expressed in terms of Lauricella hypergeometric functions of many variables. The conserved charges, i.e. the string energy, spin and angular momenta, are also found.

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