# contour integrals

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## 1—10 of 53 matching pages

##### 1: 9.14 Incomplete Airy Functions
Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …
##### 2: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
##### 3: 12.16 Mathematical Applications
PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). …
##### 4: 21.1 Special Notation
 $g,h$ positive integers. … line integral of the differential $\omega$ over the cycle $a$.
##### 5: 27.18 Methods of Computation: Primes
An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). …
##### 6: 21.7 Riemann Surfaces
21.7.5 $\oint_{a_{k}}\omega_{j}=\delta_{j,k},$ $j,k=1,2,\dots,g$.
21.7.6 $\Omega_{jk}=\oint_{b_{k}}\omega_{j},$ $j,k=1,2,\dots,g$,
##### 10: 2.4 Contour Integrals
###### §2.4 ContourIntegrals
Let $\mathscr{P}$ denote the path for the contour integral
2.4.14 $I(z)=\int_{t_{0}}^{b}e^{-zp(t)}q(t)\mathrm{d}t-\int_{t_{0}}^{a}e^{-zp(t)}q(t)% \mathrm{d}t,$
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). …