Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the nonoscillatory phase function for Legendre's differential
  equation   We express a certain complex-valued solution of Legendre's differential
equation as the product of an oscillatory exponential function and an integral
involving only nonoscillatory elementary functions. By calculating the
logarithmic derivative of this solution, we show that Legendre's differential
equation admits a nonoscillatory phase function. Moreover, we derive from our
expression an asymptotic expansion useful for evaluating Legendre functions of
the first and second kinds of large orders, as well as the derivative of the
nonoscillatory phase function. Numerical experiments demonstrating the
properties of our asymptotic expansion are presented.

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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: nlin
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The categories of this paper are: q-alg
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The categories of this paper are: bayes-an
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The categories of this paper are: patt-sol