limiting forms as trigonometric functions
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11: 28.28 Integrals, Integral Representations, and Integral Equations
12: 29.5 Special Cases and Limiting Forms
§29.5 Special Cases and Limiting Forms
… βΊ
29.5.5
even,
βΊ
29.5.6
odd,
βΊwhere is the hypergeometric function; see §15.2(i).
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βΊwhere and are Mathieu functions; see §28.2(vi).
13: 13.2 Definitions and Basic Properties
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βΊIt can be regarded as the limiting form of the hypergeometric differential equation (§15.10(i)) that is obtained on replacing by , letting , and subsequently replacing the symbol by .
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βΊ
βΊAlthough does not exist when , , many formulas containing continue to apply in their limiting form.
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βΊ
§13.2(iii) Limiting Forms as
… βΊ§13.2(iv) Limiting Forms as
…14: 19.14 Reduction of General Elliptic Integrals
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βΊIn (19.14.1)–(19.14.3) both the integrand and are assumed to be nonnegative.
Cases in which can be included by application of (19.2.10).
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βΊLegendre (1825–1832) showed that every elliptic integral can be expressed in terms of the three integrals in (19.1.2) supplemented by algebraic, logarithmic, and trigonometric functions.
…The choice among 21 transformations for final reduction to Legendre’s normal form depends on inequalities involving the limits of integration and the zeros of the cubic or quartic polynomial.
A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges.
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15: 13.14 Definitions and Basic Properties
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βΊStandard solutions are:
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βΊ
βΊAlthough does not exist when , many formulas containing continue to apply in their limiting form.
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βΊ
§13.14(iii) Limiting Forms as
… βΊ§13.14(iv) Limiting Forms as
…16: 10.52 Limiting Forms
17: 7.2 Definitions
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βΊ