By Arthur F. Nikiforov, Vasilii B. Uvarov

ISBN-10: 1475715978

ISBN-13: 9781475715972

With scholars of Physics mainly in brain, we've accrued the fabric on certain services that's most vital in mathematical physics and quan tum mechanics. we've not tried to supply the main wide collec tion attainable of knowledge approximately particular services, yet have set ourselves the duty of discovering an exposition which, according to a unified process, guarantees the potential for utilizing the idea in different normal sciences, because it professional vides an easy and potent process for the self reliant answer of difficulties that come up in perform in physics, engineering and arithmetic. For the yankee variation we've got been in a position to increase a few proofs; particularly, we have now given a brand new facts of the fundamental theorem ( 3). this is often the elemental theorem of the publication; it has now been prolonged to hide distinction equations of hypergeometric sort ( 12, 13). a number of sections were simplified and include new fabric. We think that this is often the 1st time that the speculation of classical or thogonal polynomials of a discrete variable on either uniform and nonuniform lattices has been given this sort of coherent presentation, including its numerous functions in physics.

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Additional info for Special Functions of Mathematical Physics: A Unified Introduction with Applicationsv

Sample text

The polynomials Yn (x) for which p( x) satisfies (17) are known as the classical orthogonal polynomials. They are usually considered under the auxiliary conditions that p(x) > 0 and a(x) > 0 on (a, b). These conditions are satisfied by the Jacobi polynomials P::·f3(x) for a = -1, b = 1, a > -1, {3 > -1; by the Laguerre polynomials L~(x) for a= O,b = +oo,a > -1; by the Hermite polynomials Hn(x) for a = -oo, b = +oo. We observe that in these cases the condition Am =/= An can be replaced by m =/= n.

In the disk itl < R, where R is the distance from the origin to the nearest singular point of ci> ( z, t) (for fixed z). As an example we obtain the generating function for the Legendre polynomials. In this case ~(z, t) = -1 + y'1; 4t(t + z) and consequently, by (12), ci>(z,t)=-1-1 = 1 + 2st •=e(z,t) v1 1 + 4tz + 4t2 . ) for the Legendre polynomials, we have If we replace t by -t/2, we obtain the usual form of the generating function: (13) The expansion (13) converges for iti < 1 if -1 ::; z ::; 1, since the singular points of the generating function, which are at the roots of the equation 1- 2tz + t 2 = 0, are given by t 1 ,2 = e±it,i> (cos~= z) and lie on it!

An 1 Yn(z) = () -d [crn(z)p(z)] p z zn (evidently Yn(z) = BnfJ(z)). ~ = z (it assumed that p( 8) is analytic in the region inside C). 1 27rzp(z) JM C 8-Z {~ [cr(8)t] ~ 8-Z n-0 n} d8. The interchange of summation and integration can easily be justified for sufficiently smalliti and fixed z. The geometric series in the integrand can be summed, and we obtain ()= 1 z,t 27rip(z) J c p(8)d8 8-z-cr(8)t' The denominator of the integrand has, in general, two zeros. If t -+ 0, one of the zeros tends to 8 = z.

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Special Functions of Mathematical Physics: A Unified Introduction with Applicationsv by Arthur F. Nikiforov, Vasilii B. Uvarov

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