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11: 33.4 Recurrence Relations and Derivatives

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12: 2.5 Mellin Transform Methods

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§2.5(i) Introduction

►Let $f(t)$ be a locally integrable function on $(0,\mathrm{\infty })$, that is, ${\int }_{\rho }^{T}f(t)dt$ exists for all $\rho$ and $T$ satisfying . … ►Let $f(t)$ and $h(t)$ be locally integrable on $(0,\mathrm{\infty })$ and …Also, let … ►

2.5.24 $$h_2(t)=h(t)-h_1(t).$$

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Symbols:

$h(x)$: locally integrable function

Referenced by:

§2.5(iii)

Permalink:

http://dlmf.nist.gov/2.5.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §2.5(ii), §2.5 and Ch.2

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13: 31.16 Mathematical Applications

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31.16.5 $$P_j=\frac{(ϵ-j+n)j(\beta +j-1)(\gamma +\delta +j-2)}{(\gamma +\delta +2j-3)(\gamma +\delta +2j-2)},$$

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Defines:

$P_j$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

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31.16.6 $$Q_j=-aj(j+\gamma +\delta -1)-q+\frac{(j-n)(j+\beta )(j+\gamma )(j+\gamma +\delta -1)}{(2j+\gamma +\delta )(2j+\gamma +\delta -1)}+\frac{(j+n+\gamma +\delta -1)j(j+\delta -1)(j-\beta +\gamma +\delta -1)}{(2j+\gamma +\delta -1)(2j+\gamma +\delta -2)},$$

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Defines:

$Q_j$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.16.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

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31.16.7 $$R_j=\frac{(n-j)(j+n+\gamma +\delta )(j+\gamma )(j+\delta )}{(\gamma +\delta +2j)(\gamma +\delta +2j+1)}.$$

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Defines:

$R_j$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/31.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

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14: 13.7 Asymptotic Expansions for Large Argument

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13.7.5 $$\left|{\epsilon }_n(z)\right|,{\beta }^{-1}\left|{\epsilon }_n^{\prime }(z)\right|\le 2\alpha C_n\left|\frac{{\left(a\right)}_n{\left(a-b+1\right)}_n}{n!z^{a+n}}\right|\exp \left(\frac{2\alpha \rho C_1}{|z|}\right),$$

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Defines:

${\epsilon }_n(z)$: function (locally)

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\exp z$: exponential function, $!$: factorial (as in $n!$), $n$: nonnegative integer, $z$: complex variable, $C_n$: coefficient, $\alpha$ , $\beta$ and $\rho$

Referenced by:

§13.2(i)

Permalink:

http://dlmf.nist.gov/13.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(ii), §13.7 and Ch.13

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13.7.6 $$C_n=1,\chi (n),\left(\chi (n)+\sigma {\nu }^{2}n\right){\nu }^n,$$

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Defines:

$C_n$: coefficient (locally)

Symbols:

$n$: nonnegative integer, $\sigma$ , $\nu$ and $\chi (n)$: function

Permalink:

http://dlmf.nist.gov/13.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(ii), §13.7 and Ch.13

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13.7.11 $$R_n(a,b,z)=\frac{{(-1)}^{n}2\pi z^{a-b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)}\left(\sum _{s=0}^{m-1}\frac{{\left(1-a\right)}_s{\left(b-a\right)}_s}{s!}{(-z)}^{-s}{G}_{n+2a-b-s}(z)+{\left(1-a\right)}_m{\left(b-a\right)}_m{R}_{m,n}(a,b,z)\right),$$

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Defines:

$R_n(a,b,z)$: remainder (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer, $n$: nonnegative integer, $s$: nonnegative integer, $z$: complex variable and $G_p(z)$: expansion

Referenced by:

§13.29(i)

Permalink:

http://dlmf.nist.gov/13.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(iii), §13.7 and Ch.13

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13.7.12 $$G_p(z)=\frac{{\mathrm{e}}^z}{2\pi }\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }(1-p,z).$$

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Defines:

$G_p(z)$: expansion (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(iii), §13.7 and Ch.13

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15: 31.10 Integral Equations and Representations

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31.10.1 $$W(z)={\int }_C𝒦(z,t)w(t)\rho (t)dt$$

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Defines:

$W(z)$: solution (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable, $C$: contour, $\rho (t)$: weight function and $𝒦(z,t)$: kernel

Referenced by:

§31.10(i), §31.10(i)

Permalink:

http://dlmf.nist.gov/31.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10 and Ch.31

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31.10.2 $$\rho (t)=t^{\gamma -1}{(t-1)}^{\delta -1}{(t-a)}^{ϵ-1},$$

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Defines:

$\rho (t)$: weight function (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter and $a$: complex parameter

Permalink:

http://dlmf.nist.gov/31.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10 and Ch.31

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31.10.3 $$({𝒟}_z-{𝒟}_t)𝒦=0,$$

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Defines:

$𝒦(z,t)$: kernel (locally)

Symbols:

$z$: complex variable and ${𝒟}_z$: Heun’s operator

Permalink:

http://dlmf.nist.gov/31.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10 and Ch.31

… ►Fuchs–Frobenius solutions $W_m(z)={\tilde{\kappa }}_m{z}^{-\alpha }H\mathrm{\ell }(1/a,q_m;\alpha ,\alpha -\gamma +1,\alpha -\beta +1,\delta ;1/z)$ are represented in terms of Heun functions $w_m(z)=(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$ by (31.10.1) with $W(z)=W_m(z)$, $w(z)=w_m(z)$, and with kernel chosen from … ►

31.10.14 $$\left((t-z){𝒟}_s+(z-s){𝒟}_t+(s-t){𝒟}_z\right)𝒦=0,$$

ⓘ

Defines:

$𝒦(z;s,t)$: kernel (locally)

Symbols:

$z$: complex variable and ${𝒟}_z$: Heun’s operator

Permalink:

http://dlmf.nist.gov/31.10.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(ii), §31.10 and Ch.31

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16: 32.4 Isomonodromy Problems

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32.4.4

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Defines:

$𝐀(z,\lambda )$: matrix (locally)

Symbols:

$\mathrm{i}$: imaginary unit and $z$: real

Permalink:

http://dlmf.nist.gov/32.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(ii), §32.4 and Ch.32

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32.4.5

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Defines:

$𝐁(z,\lambda )$: matrix (locally)

Symbols:

$\mathrm{i}$: imaginary unit and $z$: real

Permalink:

http://dlmf.nist.gov/32.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(ii), §32.4 and Ch.32

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32.4.6

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Defines:

$𝐀(z,\lambda )$: matrix (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $z$: real and $\alpha$: arbitrary constant

Permalink:

http://dlmf.nist.gov/32.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(iii), §32.4 and Ch.32

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32.4.7

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Defines:

$𝐁(z,\lambda )$: matrix (locally)

Symbols:

$\mathrm{i}$: imaginary unit and $z$: real

Permalink:

http://dlmf.nist.gov/32.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(iii), §32.4 and Ch.32

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32.4.8

ⓘ

Defines:

$𝐀(z,\lambda )$: matrix (locally)

Symbols:

$z$: real and ${\theta }_{\mathrm{\infty }}$: constant

Permalink:

http://dlmf.nist.gov/32.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(iv), §32.4 and Ch.32

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17: 3.7 Ordinary Differential Equations

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3.7.1 $$\frac{d^{2}w}{{dz}^2}+f(z)\frac{dw}{dz}+g(z)w=h(z),$$

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Defines:

$f(z)$: function (locally), $g(z)$: function (locally) and $h(z)$: function (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $w(z)$: function

Referenced by:

§3.7(i), §3.7(ii), §3.7(ii), §3.7(ii)

Permalink:

http://dlmf.nist.gov/3.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(i), §3.7 and Ch.3

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3.7.6

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Defines:

$𝐀(\tau ,z)$: matrix (locally) and $A_{jk}(\tau ,z)$: matrix entry (locally)

Permalink:

http://dlmf.nist.gov/3.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(ii), §3.7 and Ch.3

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3.7.7 $$𝐛(\tau ,z)=\left[\begin{array}{c}{b}_1(\tau ,z)\\ b_2(\tau ,z)\end{array}\right],$$

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Defines:

$𝐛(\tau ,z)$: vector (locally) and $b_j(\tau ,z)$: vector (locally)

Permalink:

http://dlmf.nist.gov/3.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(ii), §3.7 and Ch.3

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3.7.10

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Defines:

$𝐀(\tau ,z)$: matrix (locally)

Symbols:

${\tau }_j$: change of variable and $P$: partitioning point

Permalink:

http://dlmf.nist.gov/3.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(iii), §3.7 and Ch.3

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3.7.12 $$𝐛={[b_1({\tau }_0,z_0),b_2({\tau }_0,z_0),b_1({\tau }_1,z_1),b_2({\tau }_1,z_1),\mathrm{\dots },b_1({\tau }_{P-1},z_{P-1}),b_2({\tau }_{P-1},z_{P-1})]}^{\mathrm{T}}.$$

ⓘ

Defines:

$𝐛(\tau ,z)$: vector (locally) and $𝐛(\tau ,z)$: vector (locally)

Symbols:

${\tau }_j$: change of variable and $P$: partitioning point

Permalink:

http://dlmf.nist.gov/3.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(iii), §3.7 and Ch.3

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18: 3.8 Nonlinear Equations

… ► … ►If $\zeta$ is a simple zero, then the iteration converges locally and quadratically. … ►It converges locally and quadratically for both $ℝ$ and $ℂ$. … ►The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of $q(z)$. … ►The rule converges locally and is cubically convergent. …

19: 7.17 Inverse Error Functions

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7.17.5 $$u=-2/\ln \left(\pi x^2\ln \left(1/x\right)\right),$$

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Defines:

$u$: expansion variable (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $x$: real variable

Permalink:

http://dlmf.nist.gov/7.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.17(iii), §7.17 and Ch.7

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7.17.6 $$v=\ln \left(\ln \left(1/x\right)\right)-2+\ln \pi .$$

ⓘ

Defines:

$v$: expansion variable (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $x$: real variable

Permalink:

http://dlmf.nist.gov/7.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.17(iii), §7.17 and Ch.7

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20: 12.17 Physical Applications

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