§28.28 Integrals, Integral Representations, and Integral Equations
Contents
- §28.28(i) Equations with Elementary Kernels
- §28.28(ii) Integrals of Products with Bessel Functions
- §28.28(iii) Integrals of Products of Mathieu Functions of Noninteger Order
- §28.28(iv) Integrals of Products of Mathieu Functions of Integer Order
- §28.28(v) Compendia
§28.28(i) Equations with Elementary Kernels
Let
28.28.1 |
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Then
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28.28.3 |
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28.28.4 |
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28.28.5 |
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In (28.28.7)–(28.28.9)
the paths of integration are
given by
28.28.6 |
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where and are real constants.
28.28.7 |
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28.28.8 |
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28.28.9 |
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In (28.28.11)–(28.28.14)
28.28.10 |
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28.28.11 |
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28.28.12 |
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28.28.13 |
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28.28.14 |
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In particular, when the integrals (28.28.11),
(28.28.14) converge absolutely and uniformly in the half strip
, .
28.28.15 |
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28.28.16 |
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where the upper or lower sign is taken according as
or .
For and see §§28.4 and
28.5(i).
For details and further equations see Meixner et al. (1980, §2.1.1) and
Sips (1970).
§28.28(ii) Integrals of Products with Bessel Functions
With the notations of §28.4 for and ,
§28.14 for , and (28.23.1) for
, ,
28.28.17 |
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where and are analytic functions for
and real with
28.28.18 |
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and
28.28.19 |
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In particular, for integer and ,
28.28.20 |
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where again and ,
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28.28.22 |
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28.28.23 |
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§28.28(iii) Integrals of Products of Mathieu Functions of Noninteger Order
With the parameter suppressed we use the notation
28.28.24 |
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and assume and . Then
28.28.25 |
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28.28.26 |
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where
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28.28.28 |
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28.28.29 |
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28.28.30 |
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28.28.31 |
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28.28.32 |
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where
28.28.33 |
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Also,
28.28.34 |
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where the integral is a Cauchy principal value (§1.4(v)).
§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order
Again with the parameter suppressed, let
28.28.35 |
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Then
28.28.36 |
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28.28.37 |
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where , ; . Also,
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Let
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28.28.40 |
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Then
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28.28.42 |
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where , ; , .
Also,
28.28.43 |
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Next,
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28.28.45 |
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where , ; , . Also,
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Lastly,
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28.28.48 |
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where , ; . Also,
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§28.28(v) Compendia
See Prudnikov et al. (1990, pp. 359–368),
Gradshteyn and Ryzhik (2015, §§6.91–6.93), Sips (1970), and
Meixner et al. (1980, §2.1.1).
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