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1: 1.9 Calculus of a Complex Variable
A system of open disks around infinity is given by
1.9.35 S r = { z | z | > 1 / r } { } , 0 < r < .
2: 25.5 Integral Representations
where the integration contour is a loop around the negative real axis; it starts at , encircles the origin once in the positive direction without enclosing any of the points z = ± 2 π i , ± 4 π i , …, and returns to . …
3: 2.10 Sums and Sequences
(5.11.7) shows that the integrals around the large quarter circles vanish as n . …
4: 5.11 Asymptotic Expansions
5.11.3 Γ ( z ) = e z z z ( 2 π z ) 1 / 2 Γ ( z ) e z z z ( 2 π z ) 1 / 2 k = 0 g k z k ,
5: 25.11 Hurwitz Zeta Function
where the integration contour is a loop around the negative real axis as described for (25.5.20). …
25.11.39 k = 2 k 2 k ζ ( k + 1 , 3 4 ) = 8 G ,
As β ± with s fixed, s > 1 , … As a in the sector | ph a | π δ ( < π ) , with s ( 1 ) and δ fixed, we have the asymptotic expansion … Similarly, as a in the sector | ph a | 1 2 π δ ( < 1 2 π ) , …
6: 1.10 Functions of a Complex Variable
A function whose only singularities, other than the point at infinity, are poles is called a meromorphic function. If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles. … where N and P are respectively the numbers of zeros and poles, counting multiplicity, of f within C , and Δ C ( ph f ( z ) ) is the change in any continuous branch of ph ( f ( z ) ) as z passes once around C in the positive sense. … Branches of F ( z ) can be defined, for example, in the cut plane D obtained from by removing the real axis from 1 to and from 1 to ; see Figure 1.10.1. … The product n = 1 ( 1 + a n ) , with a n 1 for all n , converges iff n = 1 ln ( 1 + a n ) converges; and it converges absolutely iff n = 1 | a n | converges. …
7: 18.39 Applications in the Physical Sciences
allows anharmonic, or amplitude dependent, frequencies of oscillation about x e , and also escape of the particle to x = + with dissociation energy D . … where the orthogonality measure is now d r , r [ 0 , ) . Orthogonality, with measure d r for r [ 0 , ) , for fixed l normalized with measure r 2 d r , r [ 0 , ) . … For applications of Legendre polynomials in fluid dynamics to study the flow around the outside of a puff of hot gas rising through the air, see Paterson (1983). …