derivatives of the error function

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11: 4.24 Inverse Trigonometric Functions: Further Properties

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4.24.10 $$\frac{d}{dz}\mathrm{arccsc}z=\mp \frac{1}{z{(z^2-1)}^{1/2}},$$ $\mathrm{\Re }z\gtrless 0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arccsc}z$: arccosecant function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.4.57 (has an error.)

Referenced by:

§4.24(ii)

Permalink:

http://dlmf.nist.gov/4.24.E10

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See also:

Annotations for §4.24(ii), §4.24 and Ch.4

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4.24.11 $$\frac{d}{dz}\mathrm{arcsec}z=\pm \frac{1}{z{(z^2-1)}^{1/2}},$$ $\mathrm{\Re }z\gtrless 0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arcsec}z$: arcsecant function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.4.56 (has an error.)

Referenced by:

§4.24(ii)

Permalink:

http://dlmf.nist.gov/4.24.E11

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See also:

Annotations for §4.24(ii), §4.24 and Ch.4

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12: 10.17 Asymptotic Expansions for Large Argument

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§10.17(ii) Asymptotic Expansions of Derivatives

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§10.17(iii) Error Bounds for Real Argument and Order

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§10.17(iv) Error Bounds for Complex Argument and Order

… ►Corresponding error bounds for (10.17.3) and (10.17.4) are obtainable by combining (10.17.13) and (10.17.14) with (10.4.4). … ►For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).

13: 4.38 Inverse Hyperbolic Functions: Further Properties

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4.38.10 $$\frac{d}{dz}\mathrm{arccosh}z=\pm {(z^2-1)}^{-1/2},$$ $\mathrm{\Re }z\gtrless 0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.6.38 (has an error.)

Permalink:

http://dlmf.nist.gov/4.38.E10

Encodings:

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See also:

Annotations for §4.38(ii), §4.38 and Ch.4

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4.38.13 $$\frac{d}{dz}\mathrm{arcsech}z=-\frac{1}{z{(1-z^2)}^{1/2}}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arcsech}z$: inverse hyperbolic secant function and $z$: complex variable

A&S Ref:

4.6.41 (has an error.)

Permalink:

http://dlmf.nist.gov/4.38.E13

Encodings:

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See also:

Annotations for §4.38(ii), §4.38 and Ch.4

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14: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►For error bounds for (8.12.7) see Paris (2002a). … ►Lastly, a uniform approximation for $\mathrm{\Gamma }(a,ax)$ for large $a$, with error bounds, can be found in Dunster (1996a). ►For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function $\mathrm{erfc}$ see Paris (2002b) and Dunster (1996a). ►

Inverse Function

… ►These expansions involve the inverse error function $\mathrm{inverfc}\left(x\right)$ (§7.17), and are uniform with respect to $q\in [0,1]$. …

15: Bibliography E

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An upper bound for the zeros of the derivative of Bessel functions. Rend. Circ. Mat. Palermo (2) 46 (1), pp. 123–130.

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External Links:

ISSN 0009-725X, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

BibTeX

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The zeros of the complementary error function. Numer. Algorithms 49 (1-4), pp. 153–157.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§7.13(ii).

Encodings:

BibTeX

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An asymptotic expansion for the first derivative of the generalized Riemann zeta function. Math. Comp. 47 (175), pp. 347–350.

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External Links:

ISSN 0025-5718, FullText, MathReview (S. Audinarayana Moorthy), ZentralBlatt

Referenced by:

(25.11.44), (25.11.45), §25.11(xii), §25.11(xii).

Encodings:

BibTeX

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Tablicy sferoidal’nyh volnovyh funkcii i ih pervyh proizvodnyh. Tom I. Izdat. “Nauka i Tehnika”, Minsk (Russian).

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Notes:

Translated title: Tables of Spheroidal Wave Functions and Their First Derivatives.

External Links:

MathReview, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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Tablicy sferoidal’nyh volnovyh funkcii iih pervyh proizvodnyh. Tom II. Izdat. “Nauka i Tehnika”, Minsk (Russian).

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Notes:

Edited by V. I. Krylov. Translated title: Tables of Spheroidal Wave Functions and Their First Derivatives.

External Links:

MathReview

Referenced by:

4th item.

Encodings:

BibTeX

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16: Bibliography L

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Monotonicity results and inequalities for the gamma and error functions. J. Comput. Appl. Math. 23 (1), pp. 25–33.

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External Links:

ISSN 0377-0427, Document, MathReview (M. E. Cohen), ZentralBlatt

Referenced by:

§7.8.

Encodings:

BibTeX

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On the Zeros of the Derivative of Bessel Functions of Second Kind. Pubblicazioni Serie III [Publication Series III], Vol. 179, Istituto per le Applicazioni del Calcolo “Mauro Picone” (IAC), Rome.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.21(vi).

Encodings:

BibTeX

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Monotonicity of the zeros of the third derivative of Bessel functions. Methods Appl. Anal. 2 (1), pp. 103–111.

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External Links:

ISSN 1073-2772, FullText, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.21(iv).

Encodings:

BibTeX

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Monotonicity in terms of order of the zeros of the derivatives of Bessel functions. Proc. Amer. Math. Soc. 108 (2), pp. 387–389.

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External Links:

ISSN 0002-9939, FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

Encodings:

BibTeX

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The zeros of the third derivative of Bessel functions of order less than one. Methods Appl. Anal. 2 (2), pp. 147–159.

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External Links:

ISSN 1073-2772, FullText, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

Encodings:

BibTeX

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17: 13.15 Recurrence Relations and Derivatives

§13.15 Recurrence Relations and Derivatives

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§13.15(i) Recurrence Relations

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§13.15(ii) Differentiation Formulas

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13.15.15 $$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{\frac{1}{2}z}{z}^{\mu -\frac{1}{2}}{M}_{\kappa ,\mu }\left(z\right)\right)={(-1)}^n{\left(-2\mu \right)}_n{\mathrm{e}}^{\frac{1}{2}z}{z}^{\mu -\frac{1}{2}(n+1)}{M}_{\kappa -\frac{1}{2}n,\mu -\frac{1}{2}n}\left(z\right),$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E15

Encodings:

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See also:

Annotations for §13.15(ii), §13.15 and Ch.13

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13.15.26 $${\left(z\frac{d}{dz}z\right)}^n\left({\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa -1}{W}_{\kappa ,\mu }\left(z\right)\right)={(-1)}^n{\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa +n-1}{W}_{\kappa +n,\mu }\left(z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.15.E26

Encodings:

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See also:

Annotations for §13.15(ii), §13.15 and Ch.13

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18: Bibliography H

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Certain integrals that contain a probability function and degenerate hypergeometric functions. Bul. Akad. S̆tiince RSS Moldoven 1969 (2), pp. 40–47 (Russian).

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External Links:

MathReview (Wazir Hasan Abdi)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

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Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:

MathReview (M. Lawrence Glasser)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

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Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.

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External Links:

ZentralBlatt

Referenced by:

§7.18(iii), §7.18(iv), §7.18(v), §7.18(vi).

Encodings:

BibTeX

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Tables of the Modified Hankel Functions of Order One-Third and of their Derivatives. Harvard University Press, Cambridge, MA.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(xiv).

Encodings:

BibTeX

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19: 18.40 Methods of Computation

… ►Results similar to these appear in Langhoff et al. (1976) in methods developed for physics applications, and which includes treatments of systems with discontinuities in $\mu (x)$, using what is referred to as the Stieltjes derivative which may be traced back to Stieltjes, as discussed by Deltour (1968, Eq. 12). ►

Derivative Rule Approach

►An alternate, and highly efficient, approach follows from the derivative rule conjecture, see Yamani and Reinhardt (1975), and references therein, namely that … ►Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2. … ►Further, exponential convergence in $N$, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate $w(x)$ for these OP systems on $x\in [-1,1]$ and $(-\mathrm{\infty },\mathrm{\infty })$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …

20: 10.41 Asymptotic Expansions for Large Order

… ►For derivations of the results in this subsection, and also error bounds, see Olver (1997b, pp. 374–378). … … ►In the case of (10.41.13) with positive real values of $z$ the result is a consequence of the error bounds given in Olver (1997b, pp. 377–378). … ►This is a consequence of the error bounds associated with these expansions. … ►