# defined by contour integrals

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## 1—10 of 15 matching pages

##### 1: 9.14 Incomplete Airy Functions

##### 2: 1.10 Functions of a Complex Variable

###### §1.10(viii) Functions Defined by Contour Integrals

…##### 3: 2.10 Sums and Sequences

##### 4: 21.7 Riemann Surfaces

*prime form*on the corresponding compact Riemann surface $\mathrm{\Gamma}$ 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 $z(a)=z(b)$. … ►

###### Winding Number

…##### 6: Errata

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*

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.

Transform | New | Abbreviated | Old |
---|---|---|---|

Notation | Notation | Notation | |

Fourier | $\mathcal{F}\left(f\right)\left(x\right)$ | $\mathcal{F}f\left(x\right)$ | |

Fourier Cosine | ${\mathcal{F}}_{c}\left(f\right)\left(x\right)$ | ${\mathcal{F}}_{c}f\left(x\right)$ | |

Fourier Sine | ${\mathcal{F}}_{s}\left(f\right)\left(x\right)$ | ${\mathcal{F}}_{s}f\left(x\right)$ | |

Laplace | $\mathcal{L}\left(f\right)\left(s\right)$ | $\mathcal{L}f\left(s\right)$ | $\mathcal{L}(f(t);s)$ |

Mellin | $\mathcal{M}\left(f\right)\left(s\right)$ | $\mathcal{M}f\left(s\right)$ | $\mathcal{M}(f;s)$ |

Hilbert | $\mathscr{H}\left(f\right)\left(s\right)$ | $\mathscr{H}f\left(s\right)$ | $\mathscr{H}(f;s)$ |

Stieltjes | $\mathcal{S}\left(f\right)\left(s\right)$ | $\mathcal{S}f\left(s\right)$ | $\mathcal{S}(f;s)$ |

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

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 $-8.4\le \frac{x-y}{\sqrt{2}}\le 8.4$ was replaced by $-12.0\le \frac{x-y}{\sqrt{2}}\le 12.0$ and $-12.7\le \frac{x+y}{\sqrt{2}}\le 4.2$ replaced by $-18.0\le \frac{x+y}{\sqrt{2}}\le 6.0$. All plots and interactive visualizations were regenerated to improve image quality.

(a) Contour plot. | (b) Density plot. |

Figure 36.3.18: Phase of hyperbolic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi}}^{(\mathrm{H})}\left(x,y,0\right)$.

(a) Contour plot. | (b) Density plot. |

Figure 36.3.19: Phase of hyperbolic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi}}^{(\mathrm{H})}\left(x,y,1\right)$.

(a) Contour plot. | (b) Density plot. |

Figure 36.3.20: Phase of hyperbolic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi}}^{(\mathrm{H})}\left(x,y,2\right)$.

(a) Contour plot. | (b) Density plot. |

Figure 36.3.21: Phase of hyperbolic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi}}^{(\mathrm{H})}\left(x,y,3\right)$.

*Reported 2016-09-28.*

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

###### §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 ${(\mathrm{ln}t)}^{q}$, $q=1,2,\mathrm{\dots}$, in the integrand, are derived in Reid (1972), Drazin and Reid (1981, Appendix), and Baldwin (1985). … ►##### 8: 10.23 Sums

*Neumann’s polynomial*, defined by the generating function: … ►

##### 9: 9.17 Methods of Computation

###### §9.17(iii) Integral Representations

►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ 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

*Jacobi transform*is defined as ►