contour integrals

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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

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$g,h$	positive integers.
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${\oint }_a\omega$	line integral of the differential $\omega$ over the cycle $a$.

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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

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21.7.5 $${\oint }_{a_k}{\omega }_j={\delta }_{j,k},$$ $j,k=1,2,\mathrm{\dots },g$.

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $g$: positive integer, $\omega$: differential and $a_j$: cycles

Permalink:

http://dlmf.nist.gov/21.7.E5

Encodings:

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See also:

Annotations for §21.7(i), §21.7 and Ch.21

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21.7.6 $${\mathrm{\Omega }}_{jk}={\oint }_{b_k}{\omega }_j,$$ $j,k=1,2,\mathrm{\dots },g$,

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Symbols:

$g$: positive integer, $\omega$: differential and $b_j$: cycles

Permalink:

http://dlmf.nist.gov/21.7.E6

Encodings:

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See also:

Annotations for §21.7(i), §21.7 and Ch.21

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7: 8.6 Integral Representations

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§8.6(ii) Contour Integrals

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8: 12.5 Integral Representations

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§12.5(ii) Contour Integrals

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9: 11.5 Integral Representations

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§11.5(ii) Contour Integrals

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10: 2.4 Contour Integrals

§2.4 Contour Integrals

… ►Let $𝒫$ denote the path for the contour integral ►

2.4.10 $$I(z)={\int }_a^b{\mathrm{e}}^{-zp(t)}q(t)dt,$$

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Defines:

$I(z)$: contour integral (locally)

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: left endpoint, $b$: right endpoint, $p(t)$: analytic function and $q(t)$: analytic function

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E10

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Annotations for §2.4(iii), §2.4 and Ch.2

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2.4.14 $$I(z)={\int }_{t_0}^b{\mathrm{e}}^{-zp(t)}q(t)dt-{\int }_{t_0}^a{\mathrm{e}}^{-zp(t)}q(t)dt,$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: left endpoint, $b$: right endpoint, $I(z)$: contour integral, $p(t)$: analytic function and $q(t)$: analytic function

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E14

Encodings:

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See also:

Annotations for §2.4(iv), §2.4 and Ch.2

… ►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). …