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1: 31.6 Path-Multiplicative Solutions
This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the z -plane that encircles s 1 and s 2 once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor e 2 ν π i . …
2: 1.9 Calculus of a Complex Variable
If f ( z ) is continuous within and on a simple closed contour C and analytic within C , then … If f ( z ) is continuous within and on a simple closed contour C and analytic within C , and if z 0 is a point within C , then …
Winding Number
If C is a closed contour, and z 0 C , then …
3: 1.10 Functions of a Complex Variable
Let C be a simple closed contour consisting of a segment 𝐴𝐵 of the real axis and a contour in the upper half-plane joining the ends of 𝐴𝐵 . … If f ( z ) is analytic within a simple closed contour C , and continuous within and on C —except in both instances for a finite number of singularities within C —then
1.10.8 1 2 π i C f ( z ) d z = sum of the residues of  f ( z )  within  C .
1.10.10 1 2 π i C z f ( z ) f ( z ) d z = (sum of locations of zeros) (sum of locations of poles) ,
If f ( z ) and g ( z ) are analytic on and inside a simple closed contour C , and | g ( z ) | < | f ( z ) | on C , then f ( z ) and f ( z ) + g ( z ) have the same number of zeros inside C . …
4: 18.10 Integral Representations
18.10.8 p n ( x ) = g 0 ( x ) 2 π i C ( g 1 ( z , x ) ) n g 2 ( z , x ) ( z c ) 1 d z
Here C is a simple closed contour encircling z = c once in the positive sense. …
5: 3.3 Interpolation
3.3.6 R n ( z ) = ω n + 1 ( z ) 2 π i C f ( ζ ) ( ζ z ) ω n + 1 ( ζ ) d ζ ,
where C is a simple closed contour in D described in the positive rotational sense and enclosing the points z , z 1 , z 2 , , z n . … where ω n + 1 ( ζ ) is given by (3.3.3), and C is a simple closed contour in D described in the positive rotational sense and enclosing z 0 , z 1 , , z n . …
6: 3.4 Differentiation
3.4.17 1 k ! f ( k ) ( x 0 ) = 1 2 π i C f ( ζ ) ( ζ x 0 ) k + 1 d ζ ,
where C is a simple closed contour described in the positive rotational sense such that C and its interior lie in the domain of analyticity of f , and x 0 is interior to C . …
7: 2.10 Sums and Sequences
where 𝒞 is a simple closed contour in the annulus that encloses z = 0 . …
8: Mathematical Introduction
complex plane (excluding infinity).
f ( z ) | C = 0 f ( z ) is continuous at all points of a simple closed contour C in .
9: 14.25 Integral Representations
where the multivalued functions have their principal values when 1 < z < and are continuous in ( , 1 ] . For corresponding contour integrals, with less restrictions on μ and ν , see Olver (1997b, pp. 174–179), and for further integral representations see Magnus et al. (1966, §4.6.1).
10: 36.7 Zeros
Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the z -axis that is far from the origin, the zero contours form an array of rings close to the planes …