as z→0
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1—10 of 616 matching pages
1: 4.31 Special Values and Limits
2: 10.72 Mathematical Applications
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►If has a double zero , or more generally is a zero of order , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order .
The number can also be replaced by any real constant
in the sense that
is analytic and nonvanishing at ; moreover, is permitted to have a single or double pole at .
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►In regions in which the function has a simple pole at and is analytic at (the case in §10.72(i)), asymptotic expansions of the solutions of (10.72.1) for large can be constructed in terms of Bessel functions and modified Bessel functions of order , where is the limiting value of as .
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►In (10.72.1) assume and depend continuously on a real parameter , has a simple zero and a double pole , except for a critical value , where .
…These approximations are uniform with respect to both and , including , the cut neighborhood of , and .
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3: 15.15 Sums
4: 1.10 Functions of a Complex Variable
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►Then is an isolated singularity of .
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►For example, is a branch point of .
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►Furthermore, if is analytic at , then
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►Let .
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►Also, if in addition is analytic at , then
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5: 9.4 Maclaurin Series
6: 20.4 Values at = 0
7: 4.6 Power Series
8: 32.4 Isomonodromy Problems
9: 29.16 Asymptotic Expansions
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►The approximations for Lamé polynomials hold uniformly on the rectangle , , when and assume large real values.
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