Uniform Asymptotic Expansion of the Associated Legendre Function to Leading Term for Complex Degree and Integral Order

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The associated Legendre function arises naturally in the study of spherical waves. Since in practical applications it is most often symbolically represented by <formula formulatype="inline"><tex>$P_{n}^{m}(\xi)$</tex></formula> for <formula formulatype="inline"><tex>$m \leq n$</tex></formula> and <formula formulatype="inline"> <tex>$P_{n}^{m}(\xi) \equiv 0$</tex></formula> for <formula formulatype="inline"> <tex>$m&gt;n$</tex></formula> where <formula formulatype="inline"><tex>$m$</tex> </formula> is the integer order and <formula formulatype="inline"><tex>$n$</tex> </formula> is the integer degree, this form will be employed to develop the uniform asymptotic expansion. The considerable extent to which this function appears in literature substantiates its importance in engineering and science, and particularly to spherical harmonics. In his book, “<emphasis>Partial Differential Equations in Physics</emphasis>,” Sommerfeld covers a variety of subjects including spherical harmonics, and gives a detailed account of obtaining an expansion of the associated Legendre function, <formula formulatype="inline"> <tex>$P_{n}^{m}(\cos(\theta))$</tex></formula>, by the method of steepest descents over the interval <formula formulatype="inline"><tex>$0\leq\theta\leq\pi$</tex> </formula>. The results he obtains are quite accurate for <formula formulatype="inline"> <tex>$n\gg m$</tex></formula> except as <formula formulatype="inline"><tex>$\theta$</tex> </formula> approaches the critical points, <formula formulatype="inline"> <tex>$\theta\rightarrow 0$</tex></formula> or <formula formulatype="inline"> <tex>$\theta\rightarrow \pi$</tex></formula>. Beginning with the same integral representation of the associated Legendre function with integer order and degree that Sommerfeld employed, a uniform asymptotic expansion is found that is applicable to the neighborhoods of <formula formulatype="inline"><tex>$\theta=0$</tex> </formula> and <formula formulatype="inline"><tex>$\theta=\pi$</tex></formula> and that becomes increasingly more accurate as <formula formulatype="inline"> <tex>$n$</tex></formula> increases beyond <formula formulatype="inline"><tex>$m$</tex> </formula>. Furthermore, the accuracy of the resulting uniform asymptotic expansion remains for real degree and complex degree as well. The results are plotted in order to assess the accuracy and the domain of validity of the uniform asymptotic expansion. The results of the uniform asymptotic expansion are also compared to the available approximation of the associated Legendre function given in terms of Bessel functions for small values of <formula formulatype="inline"> <tex>$\theta$</tex></formula>. </para>