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11: 4.7 Derivatives and Differential Equations

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4.7.1

\frac{\mathrm{d}}{\mathrm{d}z}\ln z=\frac{1}{z},

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 4.1.46
Permalink:: http://dlmf.nist.gov/4.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §4.7(i), §4.7 and Ch.4

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4.7.5

\frac{\mathrm{d}w}{\mathrm{d}z}=\frac{f^{\prime}(z)}{f(z)}

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $f(z)$ : non-vanishing analytic function
Permalink:: http://dlmf.nist.gov/4.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §4.7(i), §4.7 and Ch.4

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4.7.7

\frac{\mathrm{d}}{\mathrm{d}z}e^{z}=e^{z},

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable
A&S Ref:: 4.2.49
Permalink:: http://dlmf.nist.gov/4.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §4.7(ii), §4.7 and Ch.4

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4.7.10

\frac{\mathrm{d}}{\mathrm{d}z}z^{a}=az^{a-1},

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $a$ : real or complex constant and $z$ : complex variable
A&S Ref:: 4.2.52
Permalink:: http://dlmf.nist.gov/4.7.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §4.7(ii), §4.7 and Ch.4

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4.7.12

\frac{\mathrm{d}w}{\mathrm{d}z}=f(z)w

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $f(z)$ : non-vanishing analytic function
Permalink:: http://dlmf.nist.gov/4.7.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §4.7(ii), §4.7 and Ch.4

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12: 27.6 Divisor Sums

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27.6.1

\sum_{d\mathbin{|}n}\lambda\left(d\right)=\begin{cases}1,&n\mbox{ is a square}% ,\\ 0,&\mbox{otherwise}.\end{cases}

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Symbols:: $\lambda\left(\NVar{n}\right)$ : Liouville’s function, $d$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §27.6 and Ch.27

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27.6.2

\sum_{d\mathbin{|}n}\mu\left(d\right)f(d)=\prod_{p\mathbin{|}n}(1-f(p)),

n>1

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, $d$ : positive integer, $n$ : positive integer and $p,p_{1},\ldots$ : prime numbers
Permalink:: http://dlmf.nist.gov/27.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §27.6 and Ch.27

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27.6.6

\sum_{d\mathbin{|}n}\phi_{k}\left(d\right)\left(\frac{n}{d}\right)^{k}=1^{k}+2% ^{k}+\dots+n^{k},

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Symbols:: $\phi_{\NVar{k}}\left(\NVar{n}\right)$ : sum of powers of integers relatively prime to a number, $d$ : positive integer, $k$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §27.6 and Ch.27

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27.6.7

\sum_{d\mathbin{|}n}\mu\left(d\right)\left(\frac{n}{d}\right)^{k}=J_{k}\left(n% \right),

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Symbols:: $J_{\NVar{k}}\left(\NVar{n}\right)$ : Jordan’s function, $\mu\left(\NVar{n}\right)$ : Möbius function, $d$ : positive integer, $k$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §27.6 and Ch.27

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27.6.8

\sum_{d\mathbin{|}n}J_{k}\left(d\right)=n^{k}.

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Symbols:: $J_{\NVar{k}}\left(\NVar{n}\right)$ : Jordan’s function, $d$ : positive integer, $k$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.6.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §27.6 and Ch.27

13: Bibliography D

Bibliography D

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G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris (1994) Detection of the density matrix through optical homodyne tomography without filtered back projection. Phys. Rev. A 50 (5), pp. 4298–4302.

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External Links:: Document
Referenced by:: §13.28(iii).
Encodings:: BibTeX

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M. D’Ocagne (1904) Sur une classe de nombres rationnels réductibles aux nombres de Bernoulli. Bull. Sci. Math. (2) 28, pp. 29–32 (French).

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External Links:: ZentralBlatt
Referenced by:: §24.1.
Encodings:: BibTeX

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P. Deligne, P. Etingof, D. S. Freed, D. Kazhdan, J. W. Morgan, and D. R. Morrison (Eds.) (1999) Quantum Fields and Strings: A Course for Mathematicians. Vol. 1, 2. American Mathematical Society, Providence, RI.

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Notes:: Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997
External Links:: ISBN 0-8218-1198-3, MathReview, ZentralBlatt
Referenced by:: §21.9.
Encodings:: BibTeX

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T. M. Dunster (1996c) Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Roderick Wong), ZentralBlatt
Referenced by:: §2.11(v).
Encodings:: BibTeX

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14: 3.2 Linear Algebra

… ►For more information on pivoting see Golub and Van Loan (1996, pp. 109–123). … ►

3.2.9

\mathbf{U}=\begin{bmatrix}d_{1}&u_{1}&&&0\\ 0&d_{2}&u_{2}&&\\ &\ddots&\ddots&\ddots&\\ &&0&d_{n-1}&u_{n-1}\\ 0&&&0&d_{n}\end{bmatrix},

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Defines:: $u_{j}$ : elements (locally)
Symbols:: $d_{j}$ : coefficient
Permalink:: http://dlmf.nist.gov/3.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §3.2(ii), §3.2 and Ch.3

►where

u_{j}=c_{j}

j=1,2,\dots,n-1

d_{1}=b_{1}

, and ►

\ell_{j}=a_{j}/d_{j-1},

… ►and back substitution is

x_{n}=y_{n}/d_{n}

, followed by …

15: 31 Heun Functions

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16: 27.5 Inversion Formulas

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27.5.1

h(n)=\sum_{d\mathbin{|}n}f(d)g\left(\frac{n}{d}\right),

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Symbols:: $d$ : positive integer, $n$ : positive integer, $f$ : function, $g$ : function and $h$ : function
Permalink:: http://dlmf.nist.gov/27.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §27.5 and Ch.27

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27.5.2

\sum_{d\mathbin{|}n}\mu\left(d\right)=\left\lfloor\frac{1}{n}\right\rfloor,

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $d$ : positive integer and $n$ : positive integer
A&S Ref:: 24.3.1 II.B (in slightly different form)
Referenced by:: §27.5
Permalink:: http://dlmf.nist.gov/27.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §27.5 and Ch.27

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27.5.3

g(n)=\sum_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\sum_{d\mathbin{|}n}g(d)% \mu\left(\frac{n}{d}\right).

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, $d$ : positive integer, $n$ : positive integer, $f$ : function and $g$ : function
A&S Ref:: 24.3.1 II.C
Permalink:: http://dlmf.nist.gov/27.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §27.5 and Ch.27

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27.5.4

n=\sum_{d\mathbin{|}n}\phi\left(d\right)\Longleftrightarrow\phi\left(n\right)=% \sum_{d\mathbin{|}n}d\mu\left(\frac{n}{d}\right),

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Symbols:: $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\mu\left(\NVar{n}\right)$ : Möbius function, $d$ : positive integer and $n$ : positive integer
A&S Ref:: 24.3.2 II.B
Permalink:: http://dlmf.nist.gov/27.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §27.5 and Ch.27

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27.5.8

g(n)=\prod_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\prod_{d\mathbin{|}n}% \left(g\left(\frac{n}{d}\right)\right)^{\mu\left(d\right)}.

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, $d$ : positive integer, $n$ : positive integer, $f$ : function and $g$ : function
A&S Ref:: 24.3.1 II.C
Permalink:: http://dlmf.nist.gov/27.5.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §27.5 and Ch.27

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17: 30.3 Eigenvalues

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30.3.4

-1<\frac{\mathrm{d}\lambda^{m}_{n}\left(\gamma^{2}\right)}{\mathrm{d}(\gamma^{% 2})}<0.

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\lambda^{\NVar{m}}_{\NVar{n}}\left(\NVar{\gamma^{2}}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.3.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §30.3(ii), §30.3 and Ch.30

… ►For values of

r_{n}^{m}

see Meixner et al. (1980, p. 109). … ►

30.3.11

\ell_{8}=2(4m^{2}-1)^{2}A+\frac{1}{16}B+\frac{1}{8}C+\frac{1}{2}D,

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Symbols:: $m$ : nonnegative integer, $\ell_{j}$ : coefficients, $A$ , $B$ , $C$ and $D$
Permalink:: http://dlmf.nist.gov/30.3.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §30.3(iv), §30.3 and Ch.30

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D=\frac{(n-m-1)(n-m)(n-m+1)(n-m+2)(n+m-1)(n+m)(n+m+1)(n+m+2)}{(2n-3)(2n-1)^{4}% (2n+1)^{2}(2n+3)^{4}(2n+5)}.

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18: 6.12 Asymptotic Expansions

… ►For these and other error bounds see Olver (1997b, pp. 109–112) with

\alpha=0

. … ►

6.12.7

R_{n}^{(\mathrm{f})}(z)=(-1)^{n}\int_{0}^{\infty}\frac{e^{-zt}t^{2n}}{t^{2}+1}% \,\mathrm{d}t,

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Defines:: $R_{n}^{(\mathrm{f})}(z)$ : remainder term (locally)
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §6.12(ii)
Permalink:: http://dlmf.nist.gov/6.12.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §6.12(ii), §6.12 and Ch.6

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6.12.8

R_{n}^{(\mathrm{g})}(z)=(-1)^{n}\int_{0}^{\infty}\frac{e^{-zt}t^{2n+1}}{t^{2}+% 1}\,\mathrm{d}t.

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Defines:: $R_{n}^{(\mathrm{g})}(z)$ : remainder term (locally)
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $z$ : complex variable and $n$ : nonnegative integer
Referenced by:: §6.12(ii)
Permalink:: http://dlmf.nist.gov/6.12.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §6.12(ii), §6.12 and Ch.6

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19: Bibliography S

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L. L. Schumaker (1981) Spline Functions: Basic Theory. John Wiley & Sons Inc., New York.

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Notes:: Pure and Applied Mathematics, A Wiley-Interscience Publication. Reprinted by Robert E. Krieger Publishing Co. Inc., 1993.
External Links:: ISBN 0-471-76475-2, MathReview (D. D. Stancu), ZentralBlatt
Referenced by:: §24.17(ii), §3.11(vi).
Encodings:: BibTeX

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B. D. Sleeman (1969) Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.

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External Links:: Document, MathReview (J. D. DePree), ZentralBlatt
Referenced by:: §31.10(ii).
Encodings:: BibTeX

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D. Sornette (1998) Multiplicative processes and power laws. Phys. Rev. E 57 (4), pp. 4811–4813.

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External Links:: Document
Referenced by:: §8.24(i).
Encodings:: BibTeX

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G. W. Spenceley and R. M. Spenceley (1947) Smithsonian Elliptic Functions Tables. Smithsonian Miscellaneous Collections, v. 109 (Publication 3863), The Smithsonian Institution, Washington, D.C..

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