# defined by contour integrals

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## 1—10 of 15 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. …
##### 3: 2.10 Sums and Sequences
The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5. …
##### 4: 21.7 Riemann Surfaces
Then the matrix defined by …is a Riemann matrix and it is used to define the corresponding Riemann theta function. … Define the holomorphic differential …Then the prime form on the corresponding compact Riemann surface $\Gamma$ is defined by … Define the operation …
##### 5: 1.9 Calculus of a Complex Variable
###### §1.9(iii) Integration
If $x(t)$ and $y(t)$ are continuous and $x^{\prime}(t)$ and $y^{\prime}(t)$ are piecewise continuous, then $z(t)$ defines a contour. … A simple closed contour is a simple contour, except that $z(a)=z(b)$. …
##### 6: Errata
• Subsection 19.2(ii) and Equation (19.2.9)

The material surrounding (19.2.8), (19.2.9) has been updated so that the complementary complete elliptic integrals of the first and second kind are defined with consistent multivalued properties and correct analytic continuation. In particular, (19.2.9) has been corrected to read

19.2.9
${K^{\prime}}\left(k\right)=\begin{cases}K\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ K\left(k^{\prime}\right)\mp 2\mathrm{i}K\left(-k\right),&\tfrac{1}{2}\pi<\pm% \operatorname{ph}k<\pi,\end{cases}$
${E^{\prime}}\left(k\right)=\begin{cases}E\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ E\left(k^{\prime}\right)\mp 2\mathrm{i}(K\left(-k\right)-E\left(-k\right)),&% \tfrac{1}{2}\pi<\pm\operatorname{ph}k<\pi\end{cases}$
• Equation (10.23.11)
10.23.11 $a_{k}=\frac{1}{2\pi i}\int_{|t|=c^{\prime}}f(t)O_{k}\left(t\right)\,\mathrm{d}t,$ $0

Originally the contour of integration written incorrectly as $|z|=c^{\prime}$, has been corrected to be $|t|=c^{\prime}$.

Reported by Mark Dunster on 2021-03-22

• Section 1.14

There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

• Figures 36.3.18, 36.3.19, 36.3.20, 36.3.21

The scaling error reported on 2016-09-12 by Dan Piponi also applied to contour and density plots for the phase of the hyperbolic umbilic canonical integrals. Scales were corrected in all figures. The interval $-8.4\leq\frac{x-y}{\sqrt{2}}\leq 8.4$ was replaced by $-12.0\leq\frac{x-y}{\sqrt{2}}\leq 12.0$ and $-12.7\leq\frac{x+y}{\sqrt{2}}\leq 4.2$ replaced by $-18.0\leq\frac{x+y}{\sqrt{2}}\leq 6.0$. All plots and interactive visualizations were regenerated to improve image quality.

Reported 2016-09-28.

• Subsections 1.15(vi), 1.15(vii), 2.6(iii)

A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order $\alpha$ was more precisely identified as the Riemann-Liouville fractional integral operator of order $\alpha$, and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).

• ##### 7: 9.13 Generalized Airy Functions
For properties of the zeros of the functions defined in this subsection see Laforgia and Muldoon (1988) and references given therein. …
###### §9.13(ii) Generalizations from Integral Representations
Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow: … Further properties of these functions, and also of similar contour integrals containing an additional factor $(\ln t)^{q}$, $q=1,2,\ldots$, in the integrand, are derived in Reid (1972), Drazin and Reid (1981, Appendix), and Baldwin (1985). …
##### 8: 10.23 Sums
Define
10.23.11 $a_{k}=\frac{1}{2\pi i}\int_{|t|=c^{\prime}}f(t)O_{k}\left(t\right)\,\mathrm{d}t,$ $0,
and $O_{k}\left(t\right)$ is Neumann’s polynomial, defined by the generating function: … and define
##### 9: 9.17 Methods of Computation
A comprehensive and powerful approach is to integrate the defining differential equation (9.2.1) by direct numerical methods. …
###### §9.17(iii) Integral Representations
Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\operatorname{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. … In the first method the integration path for the contour integral (9.5.4) is deformed to coincide with paths of steepest descent (§2.4(iv)). …The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). …
##### 10: 15.9 Relations to Other Functions
The Jacobi transform is defined as
15.9.12 $\widetilde{f}(\lambda)=\int_{0}^{\infty}f(t)\phi^{(\alpha,\beta)}_{\lambda}% \left(t\right)(2\sinh t)^{2\alpha+1}(2\cosh t)^{2\beta+1}\,\mathrm{d}t,$
where the contour of integration is located to the right of the poles of the gamma functions in the integrand, and … It is defined by: …