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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
A simple closed contour is a simple contour, except that z ( a ) = z ( b ) . … Any simple closed contour C divides into two open domains that have C as common boundary. … 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 …
3: 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 . …
4: 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 . …
5: 2.10 Sums and Sequences
where 𝒞 is a simple closed contour in the annulus that encloses z = 0 . …
6: 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 … 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 . …
7: Mathematical Introduction
complex plane (excluding infinity).
f ( z ) | C = 0 f ( z ) is continuous at all points of a simple closed contour C in .
8: 18.10 Integral Representations
Here C is a simple closed contour encircling z = c once in the positive sense. …
9: 2.4 Contour Integrals
§2.4 Contour Integrals
is seen to converge absolutely at each limit, and be independent of σ [ c , ) . … The most successful results are obtained on moving the integration contour as far to the left as possible. … Let 𝒫 denote the path for the contour integral … and assigning an appropriate value to c to modify the contour, the approximating integral is reducible to an Airy function or a Scorer function (§§9.2, 9.12). …