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1: 30.5 Functions of the Second Kind
2: 30.4 Functions of the First Kind
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►The eigenfunctions of (30.2.1) that correspond to the eigenvalues are denoted by , .
…the sign of being when is even, and the sign of being when is odd.
►When
is the prolate angular spheroidal wave function, and when
is the oblate angular spheroidal wave function.
If , reduces to the Ferrers function :
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has exactly zeros in the interval .
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3: 30.1 Special Notation
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►The main functions treated in this chapter are the eigenvalues and the spheroidal wave functions , , , , and , .
…Meixner and Schäfke (1954) use , , , for , , , , respectively.
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►Flammer (1957) and Abramowitz and Stegun (1964) use for , for , and
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4: 30.7 Graphics
5: 14.22 Graphics
6: 30.10 Series and Integrals
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►Integrals and integral equations for are given in Arscott (1964b, §8.6), Erdélyi et al. (1955, §16.13), Flammer (1957, Chapter 5), and Meixner (1951).
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7: 4.37 Inverse Hyperbolic Functions
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►These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively.
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§4.37(iv) Logarithmic Forms
… ►On the cuts … ►On the part of the cuts from to … ►On the cuts …8: 30.6 Functions of Complex Argument
9: 4.2 Definitions
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►Most texts extend the definition of the principal value to include the branch cut
…where is the excess of the number of times the path in (4.2.1) crosses the negative real axis in the positive sense over the number of times in the negative sense.
►In the DLMF we allow a further extension by regarding the cut as representing two sets of points, one set corresponding to the “upper side” and denoted by , the other set corresponding to the “lower side” and denoted by .
…Consequently is two-valued on the cut, and discontinuous across the cut.
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►This is an analytic function of on , and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless .
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