More concisely, if , then each of
(19.16.14)–(19.16.18) and (19.16.20)–(19.16.23)
satisfies Euler’s homogeneity relation:
19.18.11 |
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and also a system of Euler–Poisson differential equations (of
which only are independent):
19.18.12 |
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or equivalently,
19.18.13 |
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Here and . For group-theoretical aspects of this
system see Carlson (1963, §VI). If , then elimination of
between (19.18.11) and (19.18.12), followed by
the substitution , produces the Gauss
hypergeometric equation (15.10.1).
The function satisfies
an Euler–Poisson–Darboux equation:
19.18.14 |
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Also , with
, satisfies a wave equation:
19.18.15 |
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Similarly, the function
satisfies an equation
of axially symmetric potential theory:
19.18.16 |
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and , with
, satisfies Laplace’s equation:
19.18.17 |
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