derivatives with respect to order

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21: Bibliography S

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On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.

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External Links:

ISSN 0259-9791, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.

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External Links:

ISSN 0259-9791, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

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22: 36.11 Leading-Order Asymptotics

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36.11.2 $${\mathrm{\Psi }}_K\left(𝐱\right)=\sqrt{2\pi }\sum _{j=1}^{j_{\max }(𝐱)}\exp \left(\mathrm{i}\left({\mathrm{\Phi }}_K(t_j(𝐱);𝐱)+\frac{1}{4}\pi {(-1)}^{j+K+1}\right)\right){\left|\frac{{\partial }^2{\mathrm{\Phi }}_K(t_j(𝐱);𝐱)}{{\partial t}^2}\right|}^{-1/2}(1+o\left(1\right)).$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $t$: variable, $K$: codimension and $t_j(𝐱)$: solutions

Permalink:

http://dlmf.nist.gov/36.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11 and Ch.36

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23: 14.20 Conical (or Mehler) Functions

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14.20.1 $$\left(1-x^2\right)\frac{d^{2}w}{{dx}^2}-2x\frac{dw}{dx}-\left({\tau }^2+\frac{1}{4}+\frac{{\mu }^2}{1-x^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.20(i), §14.20(i), 2nd item

Permalink:

http://dlmf.nist.gov/14.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(i), §14.20 and Ch.14

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24: 22.13 Derivatives and Differential Equations

§22.13 Derivatives and Differential Equations

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§22.13(i) Derivatives

… ►Note that each derivative in Table 22.13.1 is a constant multiple of the product of the corresponding copolar functions. … ►

§22.13(ii) First-Order Differential Equations

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§22.13(iii) Second-Order Differential Equations

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25: 1.13 Differential Equations

… ►(More generally in (1.13.5) for $n$th-order differential equations, $f(z)$ is the coefficient multiplying the $(n-1)$th-order derivative of the solution divided by the coefficient multiplying the $n$th-order derivative of the solution, see Ince (1926, §5.2).) … ► $u$ and $z$ belong to domains $U$ and $D$ respectively, the coefficients $f(u,z)$ and $g(u,z)$ are continuous functions of both variables, and for each fixed $u$ (fixed $z$) the two functions are analytic in $z$ (in $u$). … ►Here dots denote differentiations with respect to $\zeta$, and $\{z,\zeta \}$ is the Schwarzian derivative: … ►For extensions of these results to linear homogeneous differential equations of arbitrary order see Spigler (1984). … ►This is the Sturm-Liouville form of a second order differential equation, where ^′ denotes $\frac{d}{dx}$. …

26: 10.72 Mathematical Applications

… ►These expansions are uniform with respect to $z$, including the turning point $z_0$ and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. … ►In regions in which the function $f(z)$ has a simple pole at $z=z_0$ and ${(z-z_0)}^{2}g(z)$ is analytic at $z=z_0$ (the case $\lambda =-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm \sqrt{1+4\rho }$, where $\rho$ is the limiting value of ${(z-z_0)}^{2}g(z)$ as $z\to z_0$. These asymptotic expansions are uniform with respect to $z$, including cut neighborhoods of $z_0$, and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. … ►Then for large $u$ asymptotic approximations of the solutions $w$ can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on $u$ and $\alpha$). These approximations are uniform with respect to both $z$ and $\alpha$, including $z=z_0(a)$, the cut neighborhood of $z=0$, and $\alpha =a$. …

27: 10.24 Functions of Imaginary Order

§10.24 Functions of Imaginary Order

… ►and ${\tilde{J}}_{\nu }\left(x\right)$, ${\tilde{Y}}_{\nu }\left(x\right)$ are linearly independent solutions of (10.24.1): … ► … ►In this reference ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$ are denoted respectively by $F_{\mathrm{i}\nu }(x)$ and $G_{\mathrm{i}\nu }(x)$.

28: 16.8 Differential Equations

§16.8 Differential Equations

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§16.8(i) Classification of Singularities

… ►Equation (16.8.3) is of order $\max (p,q+1)$. In Letessier et al. (1994) examples are discussed in which the generalized hypergeometric function satisfies a differential equation that is of order 1 or even 2 less than might be expected. … ►For other values of the $b_j$, series solutions in powers of $z$ (possibly involving also $\ln z$) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations. …

29: 36.12 Uniform Approximation of Integrals

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36.12.3 $$I(𝐲,k)=\frac{\exp \left(\mathrm{i}kA(𝐲)\right)}{k^{1/(K+2)}}\sum _{m=0}^K\frac{a_m(𝐲)}{k^{m/(K+2)}}\left({\delta }_{m,0}-\left(1-{\delta }_{m,0}\right)\mathrm{i}\frac{\partial }{\partial z_m}\right){\mathrm{\Psi }}_K\left(𝐳(𝐲;k)\right)\left(1+O\left(\frac{1}{k}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${\delta }_{j,k}$: Kronecker delta, ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: variable, $K$: codimension, $I(𝐲,k)$: oscillatory integral, $a_m(𝐲)$: function, $A(𝐲)$: function, $𝐳(𝐲,k)$: function and $z_m(𝐲,k)$: function

Permalink:

http://dlmf.nist.gov/36.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

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30: 1.15 Summability Methods

… ►For $\mathrm{\Re }\alpha >0$ and $x\ge 0$, the Riemann-Liouville fractional integral of order $\alpha$ is defined by … ►Also, we can replace the lower and upper limits of the integral by $x$ and $a$, respectively. … ►

§1.15(vii) Fractional Derivatives

►For , $n$ an integer, and $x\ge 0$, the fractional derivative of order $\alpha$ is defined by …Note that ${𝐷}^{1/2}𝐷\ne {𝐷}^{3/2}$. …