Search results for "Carlson"

Indefinite integrals involving the incomplete elliptic integrals of the first and second kinds

2016

ABSTRACTA substantial number of indefinite integrals are presented for the incomplete elliptic integrals of the first and second kinds. The number of new results presented is about three times the total number to be found in the current literature. These integrals were obtained with a Lagrangian method based on the differential equations which these functions obey. All results have been checked numerically with Mathematica. Similar results for the incomplete elliptic integral of the third kind will be presented separately.

Indefinite integrals involving the incomplete elliptic integral of the third kind

2016

ABSTRACTA substantial number of new indefinite integrals involving the incomplete elliptic integral of the third kind are presented, together with a few integrals for the other two kinds of incomplete elliptic integral. These have been derived using a Lagrangian method which is based on the differential equations which these functions satisfy. Techniques for obtaining new integrals are discussed, together with transformations of the governing differential equations. Integrals involving products combining elliptic integrals of different kinds are also presented.

Indefinite integrals involving complete elliptic integrals of the third kind

2017

ABSTRACTA method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete …

A Lagrangian method for deriving new indefinite integrals of special functions

2015

A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral is derived which involves an arbitrary function, and therefore yields an infinite number of indefinite integrals for any special function which obeys such a differential equation. Techniques are presented to obtain the more interesting integrals generated by such an approach, and many integrals, both previously known and completely new are derived using the method. Sample results are given for Bessel functions, Airy functions, Legendre functions and hype…

Fourier series for elliptic integrals and some generalizations via hypergeometric series

2008

Fourier series are derived for generalizations of the three canonical Legendre incomplete elliptic integrals using a hypergeometric series approach. The Fourier series for the incomplete Epstein–Hubbell integrals are obtained as special cases of the generalization of the Legendre integrals of the first and second kinds. The Fourier series for the integrals of the first and second kinds, and those for the Epstein–Hubbell integrals, were obtained recently using a different approach, but the series obtained for the generalization of the incomplete integral of the third kind is new. All cases of the integral of the third kind are given, with the modulus and the parameter being complex variables…

Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions

2017

Integration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the indep...

Indefinite integrals involving the Jacobi Zeta and Heuman Lambda functions

2017

ABSTRACTJacobian elliptic functions are used to obtain formulas for deriving indefinite integrals for the Jacobi Zeta function and Heuman's Lambda function. Only sample results are presented, mostly obtained from powers of the twelve Glaisher elliptic functions. However, this sample includes all such integrals in the literature, together with many new integrals. The method used is based on the differential equations obeyed by these functions when the independent variable is the argument u of elliptic function theory. The same method was used recently, in a companion paper, to derive similar integrals for the three canonical incomplete elliptic integrals.

Lajiristeyksiä maan ääressä : lajityyppien intertekstuaalinen analyysi Kristina Carlsonin romaanista Maan ääreen

2014

Pro gradu -tutkielmani aineena on lajityyppien intertekstuaalinen analyysi Kristina Carlsonin romaanista Maan ääreen (1999). Tarkastelen romaania ja siitä kirjoitettua kymmentä kirjallisuusarvostelua, joiden mukaan teoksesta on löydettävissä useita eri genrejä. Selvitän, ilmenevätkö kirjallisuusarvosteluissa esitetyt lajit Maan ääreen -romaanissa, ja jos ilmenevät, niin millä tavoin. Tutkin myös minkälaista lisätietoa romaanin lajipiirteiden tarkastelu tuotaa teoksesta ja siitä kirjoitetuista kirjallisuusarvosteluista. Tarkastelen Maan ääreen -romaanin lajeja intertekstuaalisuuden näkökulmasta. Laajassa kontekstissa intertekstuaalisuus ilmenee muun muassa lajien välisenä kommunikaationa, jo…

A theorem of insertion and extension of functions for normal spaces

1993

On Carlson"s and Shafer"s inequalities

2014

In this paper the authors re ne the Carlson"s inequalities for inverse cosine function, and the Shafer"s inequalities for inverse tangent function.