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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Parabolic cylinder function ： ウィキペディア英語版 Parabolic cylinder function In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation :$\backslash frac\; +\; \backslash left(\backslash tildez^2+\backslash tildez+\backslash tilde\backslash right)f=0.$ This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabolic cylindrical coordinates. The above equation may be brought into two distinct forms (A) and (B) by completing the square and rescaling ''z'', called H. F. Weber's equations : :$\backslash frac\; -\; \backslash left(\backslash tfrac14z^2+a\backslash right)f=0$    (A) and :$\backslash frac\; +\; \backslash left(\backslash tfrac14z^2-a\backslash right)f=0.$     (B) If :$f(a,z)\backslash ,$ is a solution, then so are :$f(a,-z),\; f(-a,iz)\backslash textf(-a,-iz).\backslash ,$ If :$f(a,z)\backslash ,$ is a solution of equation (A), then :$f(-ia,ze^)\backslash ,$ is a solution of (B), and, by symmetry, :$f(-ia,-ze^),\; f(ia,-ze^)\backslash textf(ia,ze^)\backslash ,$ are also solutions of (B). ==Solutions== There are independent even and odd solutions of the form (A). These are given by (following the notation of Abramowitz and Stegun (1965)): :$y\_1(a;z)\; =\; \backslash exp(-z^2/4)\; \backslash ;\_1F\_1$ \left(\tfrac12a+\tfrac14; \; \tfrac12\; ; \; \frac\right)\,\,\,\,\,\, (\mathrm) and :$y\_2(a;z)\; =\; z\backslash exp(-z^2/4)\; \backslash ;\_1F\_1$ \left(\tfrac12a+\tfrac34; \; \tfrac32\; ; \; \frac\right)\,\,\,\,\,\, (\mathrm) where $\backslash ;\_1F\_1\; (a;b;z)=M(a;b;z)$ is the confluent hypergeometric function. Other pairs of independent solutions may be formed from linear combinations of the above solutions (see Abramowitz and Stegun). One such pair is based upon their behavior at infinity: :$$ U(a,z)=\frac\sin(\xi\pi)\Gamma(1-\xi)\,y_2(a,z) \right" TITLE=" \cos(\xi\pi)\Gamma(1/2-\xi)\,y_1(a,z) -\sqrt\sin(\xi\pi)\Gamma(1-\xi)\,y_2(a,z) \right">) :$$ V(a,z)=\frac \left() where :$$ \xi=\fraca+\frac . The function ''U''(''a'', ''z'') approaches zero for large values of z  and |arg(''z'')| < π/2, while ''V''(''a'', ''z'') diverges for large values of positive real ''z'' . :$$ \lim_U(a,z)/e^z^=1\,\,\,\,(\text\,|\arg(z)|<\pi/2) and :$$ \lim_V(a,z)/\sqrt}e^z^=1\,\,\,\,(\text\,\arg(z)=0) . For half-integer values of ''a'', these (that is, ''U'' and ''V'') can be re-expressed in terms of Hermite polynomials; alternatively, they can also be expressed in terms of Bessel functions. The functions ''U'' and ''V'' can also be related to the functions ''Dp''(''x'') (a notation dating back to Whittaker (1902)) that are themselves sometimes called parabolic cylinder functions (see Abramowitz and Stegun (1965)): :$U(a,x)=D\_(x),$ :$V(a,x)=\backslash frac(\backslash pi\; a)\; D\_(x)+D\_(-x))\; .$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Parabolic cylinder function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.