西武芸術科学大学商科文凭证书【仿证 微kaa77788】】uppere

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1: Bibliography Q

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“Best possible” upper and lower bounds for the zeros of the Bessel function $J_{\nu }(x)$ . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.

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External Links:

ISSN 0002-9947, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

BibTeX

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2: Sidebar 5.SB1: Gamma & Digamma Phase Plots

… ►In the upper half of the image, the poles of $\mathrm{\Gamma }\left(z\right)$ are clearly visible at negative integer values of $z$: the phase changes by $2\pi$ around each pole, showing a full revolution of the color wheel. …

3: 19.32 Conformal Map onto a Rectangle

… ►then $z(p)$ is a Schwartz–Christoffel mapping of the open upper-half $p$-plane onto the interior of the rectangle in the $z$-plane with vertices …As $p$ proceeds along the entire real axis with the upper half-plane on the right, $z$ describes the rectangle in the clockwise direction; hence $z(x_3)$ is negative imaginary. …

4: 15.17 Mathematical Applications

… ►The quotient of two solutions of (15.10.1) maps the closed upper half-plane $\mathrm{\Im }z\ge 0$ conformally onto a curvilinear triangle. …

5: 18.14 Inequalities

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§18.14(i) Upper Bounds

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Jacobi

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Ultraspherical

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Laguerre

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Hermite

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6: 4.37 Inverse Hyperbolic Functions

… ►the upper/lower signs corresponding to the right/left sides. … ►the upper or lower sign being taken according as $\mathrm{\Re }z\gtrless 0$; compare Figure 4.37.1(ii). … ►the upper/lower sign corresponding to the upper/lower side. … ►the upper/lower sign corresponding to the upper/lower side. … ►the upper/lower sign corresponding to the upper/lower sides. …

7: 25.15 Dirichlet $L$-functions

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25.15.3 $$L(s,\chi )=k^{-s}\sum _{r=1}^{k-1}\chi (r)\zeta (s,\frac{r}{k}),$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

finite sum

Source:

Apostol (1976, (8), p. 255)

Notes:

In the original source the upper-index of the finite sum is $k$. We use $k-1$ since $\chi (k)=0$.

Referenced by:

(25.11.36), (25.11.36), §25.11(x), §25.15(i), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

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25.15.6 $$G(\chi )\equiv \sum _{r=1}^{k-1}\chi (r){\mathrm{e}}^{2\pi \mathrm{i}r/k}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer and $\chi (n)$: Dirichlet character

Keywords:

definition

Source:

Apostol (1976, p. 262)

Referenced by:

Erratum (V1.1.4) for Equation (25.15.6)

Permalink:

http://dlmf.nist.gov/25.15.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.15(i), §25.15 and Ch.25

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25.15.10

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $k$: nonnegative integer and $\chi (n)$: Dirichlet character

Keywords:

zeros

Source:

Apostol (1976, Theorem 12.20, p. 268)

Referenced by:

Erratum (V1.1.4) for Equation (25.15.10)

Permalink:

http://dlmf.nist.gov/25.15.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.15(ii), §25.15 and Ch.25

8: 7.8 Inequalities

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9: 19.9 Inequalities

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19.9.17 $$L\le F(\varphi ,k)\le \sqrt{UL}\le \frac{1}{2}(U+L)\le U,$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $L$: lower limit and $U$: upper limit

Permalink:

http://dlmf.nist.gov/19.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

►where …

10: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

… ►The upper signs correspond to ${\mathrm{M}}_{\nu }^{(3)}(z,h)$ and the lower signs to ${\mathrm{M}}_{\nu }^{(4)}(z,h)$. …