The Harmonic Functions Associated with the Parabolic Cylinder
The Harmonic Functions Associated with the Parabolic Cylinder
复制标题
与抛物柱面相关的调和函数
DOI：
10.1112/plms/s2-17.1.116
复制
发表时间：
2014
影响因子：
1.8
通讯作者：
G. N. Watson
中科院分区：
文献类型：
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作者：
G. N. Watson
Dn(X)=2 h-9-其中w*(X)=and|S？(ao|<；3*35...，当n&>2；和ic，(0)=0或Gb&/»(其中-1<；9<；1)时，根据n为奇数或偶数。在我以前的文章中，我指出，当z是复数时，Adamoff的研究将推广到BLh{z)，只要[L(Z)]不太大：并且这个结果使得给出Hermite和Whittaker关于形式2的任意解析函数的可展性的一个定理的形式证明变得容易：-一个类似于Neumann在Legendre多项式和Bessel函数级数展开式中的结果.本文用最陡下降法研究了Dn(Z)的各种渐近公式和展开式；因此，文中所包含的结果将与Debite和我自己关于Bessel函数的先前结果相联系。
Dn (x)= 2h-nine-where w*(x)= and| S «(aO|< 3* 35..., when n> 2; and ic,,(0)= 0 or£ &/»(where—1< 9< 1), according as n is odd or even. In my former paper, I shewed that Adamoffs investigation would be extended to Blh {z), when z is complex, provided that\l (z)\is not too large: and this result made it easy to give a formal proof of a theorem enunciated by Hermite and Whittaker, concerning the expansibility of an arbitrary analytic function in the form 2 anDn (,?):—a result analogous to Neumann's expansions in series of Legendre polynomials and Bessel functions.In this paper, I investigate various types of asymptotic formulae and expansions of Dn (z), hy the method of steepest descents; the results contained in the paper are therefore to be associated with those previously obtained by Debyet and by myself I for Bessel functions.