高中毕业证等级h《做证微fuk7778》100

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11: 27.14 Unrestricted Partitions

… ►For example, $p\left(10\right)=42,p\left(100\right)$ = $\mathrm{1905\hspace{0.33em}69292}$, and $p\left(200\right)=\mathrm{397\hspace{0.33em}29990\hspace{0.33em}29388}$. …and $s(h,k)$ is a Dedekind sum given by … ►Dedekind sums occur in the transformation theory of the Dedekind modular function $\eta \left(\tau \right)$, defined by …$\eta \left(\tau \right)$ satisfies the following functional equation: if $a,b,c,d$ are integers with $ad-bc=1$ and $c>0$, then … ►For further properties of the function $\eta \left(\tau \right)$ see §§23.15–23.19. …

12: 28.21 Graphics

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Figure 28.21.1: ${\mathrm{Mc}}_0^{(1)}(x,h)$ for $0\le h\le 3$, $0\le x\le 2$. Magnify 3D Help

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Figure 28.21.2: ${\mathrm{Mc}}_1^{(1)}(x,h)$ for $0\le h\le 3$, $0\le x\le 2$. Magnify 3D Help

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Figure 28.21.3: ${\mathrm{Mc}}_0^{(2)}(x,h)$ for $0.1\le h\le 2$, $0\le x\le 2$. Magnify 3D Help

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Figure 28.21.4: ${\mathrm{Mc}}_1^{(2)}(x,h)$ for $0.2\le h\le 2$, $0\le x\le 2$. Magnify 3D Help

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Figure 28.21.5: ${\mathrm{Ms}}_1^{(1)}(x,h)$ for $0\le h\le 3$, $0\le x\le 2$. Magnify 3D Help

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13: 7.23 Tables

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Finn and Mugglestone (1965) includes the Voigt function $H(a,u)$, $u\in [0,22]$, $a\in [0,1]$, 6S.

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Zhang and Jin (1996, pp. 637, 639) includes $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $\mathrm{erf}x$, $x=0(.02)1(.04)3$, 8D; $C\left(x\right)$, $S\left(x\right)$, $x=0(.2)10(2)100(100)500$, 8D.

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Fettis et al. (1973) gives the first 100 zeros of $\mathrm{erf}z$ and $w\left(z\right)$ (the table on page 406 of this reference is for $w\left(z\right)$, not for $\mathrm{erfc}z$), 11S.

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14: 10.5 Wronskians and Cross-Products

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10.5.3 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(1)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(1)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(1)}\left(z\right)=2\mathrm{i}/(\pi z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

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10.5.4 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(2)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-2\mathrm{i}/(\pi z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

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10.5.5 $$𝒲\left\{H_{\nu }^{(1)}\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=H_{\nu +1}^{(1)}\left(z\right)H_{\nu }^{(2)}\left(z\right)-H_{\nu }^{(1)}\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-4\mathrm{i}/(\pi z).$$

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.17

Permalink:

http://dlmf.nist.gov/10.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

15: 26.12 Plane Partitions

… ►An equivalent definition is that a plane partition is a finite subset of $\mathbb{N}\times \mathbb{N}\times \mathbb{N}$ with the property that if $(r,s,t)\in \pi$ and $(1,1,1)\le (h,j,k)\le (r,s,t)$, then $(h,j,k)$ must be an element of $\pi$. Here $(h,j,k)\le (r,s,t)$ means $h\le r$, $j\le s$, and $k\le t$. It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point $(h,j,k)\in \pi$. … ►A plane partition is symmetric if $(h,j,k)\in \pi$ implies that $(j,h,k)\in \pi$. … ►A plane partition is cyclically symmetric if $(h,j,k)\in \pi$ implies $(j,k,h)\in \pi$. …

16: 10.11 Analytic Continuation

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$$H_{\nu }^{(1)}\left(z{\mathrm{e}}^{\pi \mathrm{i}}\right)=-{\mathrm{e}}^{-\nu \pi \mathrm{i}}{H}_{\nu }^{(2)}\left(z\right),$$

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$$H_{\nu }^{(2)}\left(z{\mathrm{e}}^{-\pi \mathrm{i}}\right)=-{\mathrm{e}}^{\nu \pi \mathrm{i}}{H}_{\nu }^{(1)}\left(z\right).$$

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10.11.7 $$H_n^{(1)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn-1}((m-1)H_n^{(1)}\left(z\right)+m{H}_n^{(2)}\left(z\right)),$$

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer and $z$: complex variable

Referenced by:

§10.17(i)

Permalink:

http://dlmf.nist.gov/10.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.11 and Ch.10

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10.11.8 $$H_n^{(2)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn}(m{H}_n^{(1)}\left(z\right)+(m+1)H_n^{(2)}\left(z\right)).$$

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer and $z$: complex variable

Referenced by:

§10.11, §10.17(i)

Permalink:

http://dlmf.nist.gov/10.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.11 and Ch.10

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$H_{\nu }^{(1)}\left(\overline{z}\right)=\overline{H_{\nu }^{(2)}\left(z\right)},$ $H_{\nu }^{(2)}\left(\overline{z}\right)=\overline{H_{\nu }^{(1)}\left(z\right)}.$

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17: 10.1 Special Notation

… ►The main functions treated in this chapter are the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$; Hankel functions $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$; modified Bessel functions $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$; spherical Bessel functions ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, ${𝗄}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. … ►Abramowitz and Stegun (1964): $j_n(z)$, $y_n(z)$, $h_n^{(1)}(z)$, $h_n^{(2)}(z)$, for ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$, respectively, when $n\ge 0$. ►Jeffreys and Jeffreys (1956): ${\mathrm{Hs}}_{\nu }(z)$ for $H_{\nu }^{(1)}\left(z\right)$, ${\mathrm{Hi}}_{\nu }(z)$ for $H_{\nu }^{(2)}\left(z\right)$, ${\mathrm{Kh}}_{\nu }(z)$ for $(2/\pi )K_{\nu }\left(z\right)$. …

18: 28.16 Asymptotic Expansions for Large $q$

… ►Then as $h(=\sqrt{q})\to +\mathrm{\infty }$ ►

28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.16 and Ch.28

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19: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

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28.24.2 $${\epsilon }_s{\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}\frac{A_{2\mathrm{\ell }}^{2m}(h^2)}{A_{2s}^{2m}(h^2)}\left(J_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }+s}^{(j)}(h{\mathrm{e}}^z)+J_{\mathrm{\ell }+s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }-s}^{(j)}(h{\mathrm{e}}^z)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, ${\epsilon }_s$: constants and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.7

Permalink:

http://dlmf.nist.gov/28.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

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28.24.3 $${\mathrm{Mc}}_{2m+1}^{(j)}(z,h)={(-1)}^m\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}\frac{A_{2\mathrm{\ell }+1}^{2m+1}(h^2)}{A_{2s+1}^{2m+1}(h^2)}\left(J_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }+s+1}^{(j)}(h{\mathrm{e}}^z)+J_{\mathrm{\ell }+s+1}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }-s}^{(j)}(h{\mathrm{e}}^z)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.8

Permalink:

http://dlmf.nist.gov/28.24.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

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28.24.4 $${\mathrm{Ms}}_{2m+1}^{(j)}(z,h)={(-1)}^m\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}\frac{B_{2\mathrm{\ell }+1}^{2m+1}(h^2)}{B_{2s+1}^{2m+1}(h^2)}\left(J_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }+s+1}^{(j)}(h{\mathrm{e}}^z)-J_{\mathrm{\ell }+s+1}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }-s}^{(j)}(h{\mathrm{e}}^z)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

A&S Ref:

20.6.9

Permalink:

http://dlmf.nist.gov/28.24.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

►

28.24.5 $${\mathrm{Ms}}_{2m+2}^{(j)}(z,h)={(-1)}^m\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}\frac{B_{2\mathrm{\ell }+2}^{2m+2}(h^2)}{B_{2s+2}^{2m+2}(h^2)}\left(J_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }+s+2}^{(j)}(h{\mathrm{e}}^z)-J_{\mathrm{\ell }+s+2}\left(h{\mathrm{e}}^{-z}\right){𝒞}_{\mathrm{\ell }-s}^{(j)}(h{\mathrm{e}}^z)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

A&S Ref:

20.6.10

Permalink:

http://dlmf.nist.gov/28.24.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

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28.24.6 $${\epsilon }_s{\mathrm{Ie}}_{2m}(z,h)={(-1)}^s\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}\frac{A_{2\mathrm{\ell }}^{2m}(h^2)}{A_{2s}^{2m}(h^2)}\left(I_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right)I_{\mathrm{\ell }+s}\left(h{\mathrm{e}}^z\right)+I_{\mathrm{\ell }+s}\left(h{\mathrm{e}}^{-z}\right)I_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^z\right)\right),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${\mathrm{Ie}}_n(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, ${\epsilon }_s$: constants and $A_m(q)$: Fourier coefficient

A&S Ref:

20.8.8

Permalink:

http://dlmf.nist.gov/28.24.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

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20: 28.23 Expansions in Series of Bessel Functions

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${𝒞}_{\mu }^{(3)}=H_{\mu }^{(1)},$

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${𝒞}_{\mu }^{(4)}=H_{\mu }^{(2)};$

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28.23.2 $${\mathrm{me}}_{\nu }(0,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{c}_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.6 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.12

Referenced by:

§28.23

Permalink:

http://dlmf.nist.gov/28.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.7 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(\frac{1}{2}\pi ,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\sinh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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