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1: 7.25 Software
2: 6.21 Software
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§6.21(iii) , , , , ,
…3: 9.20 Software
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§9.20(iii) , , , ,
…4: 5.24 Software
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§5.24(iv) , , ,
…5: 5.21 Methods of Computation
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►An effective way of computing
in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3).
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6: 10.42 Zeros
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►For example, if is real, then the zeros of are all complex unless for some positive integer , in which event has two real zeros.
►The distribution of the zeros of
in the sector
in the cases is obtained on rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively, through an angle so that in each case the cut lies along the positive imaginary axis.
The zeros in the sector are their conjugates.
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has no zeros in the sector ; this result remains true when is replaced by any real number .
For the number of zeros of
in the sector , when is real, see Watson (1944, pp. 511–513).
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7: 22.10 Maclaurin Series
8: 9.17 Methods of Computation
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►However, in the case of and this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v).
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►In the case of , for example, this means that in the sectors we may integrate along outward rays from the origin with initial values obtained from §9.2(ii).
…On the remaining rays, given by and , integration can proceed in either direction.
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►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing
in the complex plane, once values of this function can be generated on the positive real axis.
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►Gil et al. (2002c) describes two methods for the computation of and for .
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9: Software Index
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5.24(iv) , , , | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||||
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7.25(iii) , , , | ✓ | ✓ | a | ✓ | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||
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7.25(v) , , | ✓ | a | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||
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7.25(vii) , , | ✓ | ✓ | ✓ | ||||||||||||||||||||||
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9.20(iii) , , , , | ✓ | ✓ | ✓ | ✓ | a | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||||||||||
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10: 1.10 Functions of a Complex Variable
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►If is harmonic in
, , and for all , then is constant in
.
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►Suppose is multivalued and is a point such that there exists a branch of
in a cut neighborhood of , but there does not exist a branch of
in any punctured neighborhood of .
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►For each , is analytic in
; is a continuous function of both variables when and ; the integral (1.10.18) converges at , and this convergence is uniform with respect to
in every compact subset of .
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►Suppose , , a domain.
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►If there is an , independent of , such that
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