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1: Sidebar 5.SB1: Gamma & Digamma Phase Plots
In the upper half of the image, the poles of Γ ( z ) are clearly visible at negative integer values of z : the phase changes by 2 π around each pole, showing a full revolution of the color wheel. … In the lower half of the image, the poles of ψ ( z ) (corresponding to the poles of Γ ( z ) ) and the zeros between them are clear. Phase changes around the zeros are of opposite sign to those around the poles. …
2: 10.72 Mathematical Applications
The number m can also be replaced by any real constant λ ( > - 2 ) in the sense that ( z - z 0 ) - λ f ( z ) is analytic and nonvanishing at z 0 ; moreover, g ( z ) is permitted to have a single or double pole at z 0 . The order of the approximating Bessel functions, or modified Bessel functions, is 1 / ( λ + 2 ) , except in the case when g ( z ) has a double pole at z 0 . …
§10.72(ii) Differential Equations with Poles
§10.72(iii) Differential Equations with a Double Pole and a Movable Turning Point
In (10.72.1) assume f ( z ) = f ( z , α ) and g ( z ) = g ( z , α ) depend continuously on a real parameter α , f ( z , α ) has a simple zero z = z 0 ( α ) and a double pole z = 0 , except for a critical value α = a , where z 0 ( a ) = 0 . …
3: 22.4 Periods, Poles, and Zeros
§22.4 Periods, Poles, and Zeros
§22.4(i) Distribution
For each Jacobian function, Table 22.4.1 gives its periods in the z -plane in the left column, and the position of one of its poles in the second row. … The other poles and zeros are at the congruent points. … Using the p,q notation of (22.2.10), Figure 22.4.2 serves as a mnemonic for the poles, zeros, periods, and half-periods of the 12 Jacobian elliptic functions as follows. …
4: 4.28 Definitions and Periodicity
Periodicity and Zeros
5: 13.27 Mathematical Applications
For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i). …
6: 8.6 Integral Representations
where the integration path passes above or below the pole at t = 1 , according as upper or lower signs are taken. … In (8.6.10)–(8.6.12), c is a real constant and the path of integration is indented (if necessary) so that in the case of (8.6.10) it separates the poles of the gamma function from the pole at s = a , in the case of (8.6.11) it is to the right of all poles, and in the case of (8.6.12) it separates the poles of the gamma function from the poles at s = 0 , 1 , 2 , . …
7: 1.10 Functions of a Complex Variable
Lastly, if a n 0 for infinitely many negative n , then z 0 is an isolated essential singularity. … 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. … If the singularities within C are poles and f ( z ) is analytic and nonvanishing on C , then … each location again being counted with multiplicity equal to that of the corresponding zero or pole. …
8: 16.17 Definition
where the integration path L separates the poles of the factors Γ ( b - s ) from those of the factors Γ ( 1 - a + s ) . …
  • (ii)

    L is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the Γ ( b - s ) once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all z ( 0 ) if p < q , and for 0 < | z | < 1 if p = q 1 .

  • (iii)

    L is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the Γ ( 1 - a + s ) once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all z if p > q , and for | z | > 1 if p = q 1 .

  • 9: 5.2 Definitions
    It is a meromorphic function with no zeros, and with simple poles of residue ( - 1 ) n / n ! at z = - n . … ψ ( z ) is meromorphic with simple poles of residue - 1 at z = - n . …
    10: 32.11 Asymptotic Approximations for Real Variables
    Next, for given initial conditions w ( 0 ) = 0 and w ( 0 ) = k , with k real, w ( x ) has at least one pole on the real axis. … If | k | > 1 , then w k ( x ) has a pole at a finite point x = c 0 , dependent on k , and … then w h ( x ) has no poles on the real axis. … and w h * ( x ) has no poles on the real axis. Lastly if h > h * , then w h ( x ) has a simple pole on the real axis, whose location is dependent on h . …