怎么够买北加州大学文凭毕业证【仿证 微fuk7778】acoshS

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1: 4.37 Inverse Hyperbolic Functions

… ►

4.37.2 $$\mathrm{Arccosh}z={\int }_1^z\frac{dt}{{(t^2-1)}^{1/2}},$$

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Defines:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function

Symbols:

$dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Referenced by:

§4.37(i)

Permalink:

http://dlmf.nist.gov/4.37.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(i), §4.37 and Ch.4

… ►The principal branches are denoted by $\mathrm{arcsinh}$, $\mathrm{arccosh}$, $\mathrm{arctanh}$ respectively. Each is two-valued on the corresponding cut(s), and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts. … ►

4.37.11 $$\mathrm{arccosh}\left(-z\right)=\pm \pi \mathrm{i}+\mathrm{arccosh}z,$$ $\mathrm{\Im }z\gtrless 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.6.12 (has an error even in the tenth printing.)

Referenced by:

§4.37(iii)

Permalink:

http://dlmf.nist.gov/4.37.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iii), §4.37 and Ch.4

… ►

4.37.30 $$w=\mathrm{Arccosh}z=\pm \mathrm{arccosh}z+2k\pi \mathrm{i},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\mathrm{Arccosh}z$: general inverse hyperbolic cosine function, $k$: integer and $z$: complex variable

A&S Ref:

4.6.9

Permalink:

http://dlmf.nist.gov/4.37.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(v), §4.37 and Ch.4

…

2: 4.29 Graphics

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►►

Figure 4.29.2: Principal values of $\mathrm{arcsinh}x$ and $\mathrm{arccosh}x$. ($\mathrm{arccosh}x$ is complex when .) Magnify

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3: 4.38 Inverse Hyperbolic Functions: Further Properties

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4.38.3 $$\mathrm{arccosh}z=\ln \left(2z\right)-\frac{1}{2}\frac{1}{2z^2}-\frac{1\cdot 3}{2\cdot 4}\frac{1}{4z^4}-\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{1}{6z^6}-\mathrm{\cdots },$$ $|z|>1$.

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.6.32

Referenced by:

§4.38(i)

Permalink:

http://dlmf.nist.gov/4.38.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(i), §4.38 and Ch.4

►

4.38.4 $$\mathrm{arccosh}z={(2(z-1))}^{1/2}\left(1+\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{1\cdot 3\cdot 5\mathrm{\cdots }(2n-1)}{2^{2n}n!(2n+1)}{(z-1)}^n\right),$$ $\mathrm{\Re }z>0$, $|z-1|\le 2$.

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Symbols:

$!$: factorial (as in $n!$), $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Re }$: real part, $n$: integer and $z$: complex variable

Referenced by:

§4.38(i)

Permalink:

http://dlmf.nist.gov/4.38.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(i), §4.38 and Ch.4

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4.38.10 $$\frac{d}{dz}\mathrm{arccosh}z=\pm {(z^2-1)}^{-1/2},$$ $\mathrm{\Re }z\gtrless 0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.6.38 (has an error.)

Permalink:

http://dlmf.nist.gov/4.38.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(ii), §4.38 and Ch.4

… ►

4.38.16 $$\mathrm{Arccosh}u\pm \mathrm{Arccosh}v=\mathrm{Arccosh}\left(uv\pm {((u^2-1)(v^2-1))}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function

A&S Ref:

4.6.27

Permalink:

http://dlmf.nist.gov/4.38.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

… ►

4.38.18 $$\mathrm{Arcsinh}u\pm \mathrm{Arccosh}v=\mathrm{Arcsinh}\left(uv\pm {((1+u^2)(v^2-1))}^{1/2}\right)=\mathrm{Arccosh}\left(v{(1+u^2)}^{1/2}\pm u{(v^2-1)}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function and $\mathrm{Arcsinh}z$: general inverse hyperbolic sine function

A&S Ref:

4.6.29

Permalink:

http://dlmf.nist.gov/4.38.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

…

4: 4.47 Approximations

… ►Hart et al. (1968) give $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arccos$, $\arctan$, $\sinh$, $\cosh$, $\tanh$, $\mathrm{arcsinh}$, $\mathrm{arccosh}$. …

5: 19.10 Relations to Other Functions

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$$\mathrm{arccosh}\left(x/y\right)={(x^2-y^2)}^{1/2}{R}_C(x^2,y^2).$$

…

6: 4.40 Integrals

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4.40.12 $$\int \mathrm{arccosh}xdx=x\mathrm{arccosh}x-{(x^2-1)}^{1/2},$$ ,

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Symbols:

$dx$: differential of $x$ , $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\int$: integral and $x$: real variable

A&S Ref:

4.6.44 (modified)

Permalink:

http://dlmf.nist.gov/4.40.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iv), §4.40 and Ch.4

…

7: 15.12 Asymptotic Approximations

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15.12.6 $$\zeta =\mathrm{arccosh}z.$$

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $z$: complex variable and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/15.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

… ►

15.12.10 $$\zeta =\mathrm{arccosh}\left(\frac{1}{4}z-1\right),$$

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $z$: complex variable and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/15.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

…

8: 13.20 Uniform Asymptotic Approximations for Large $\mu$

… ►

13.20.13 $$\zeta \sqrt{{\zeta }^2-{\alpha }^2}-{\alpha }^2\mathrm{arccosh}\left(\frac{\zeta }{\alpha }\right)=\frac{X}{\mu }-\frac{2\kappa }{\mu }\ln \left(\frac{X+x-2\kappa }{2\sqrt{{\kappa }^2-{\mu }^2}}\right)-2\ln \left(\frac{\kappa x-\mu X-2{\mu }^2}{x\sqrt{{\kappa }^2-{\mu }^2}}\right),$$ $x\ge 2\kappa +2\sqrt{{\kappa }^2-{\mu }^2}$,

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

… ►

13.20.15 $$-\zeta \sqrt{{\zeta }^2-{\alpha }^2}-{\alpha }^2\mathrm{arccosh}\left(-\frac{\zeta }{\alpha }\right)=-\frac{X}{\mu }+\frac{2\kappa }{\mu }\ln \left(\frac{2\kappa -X-x}{2\sqrt{{\kappa }^2-{\mu }^2}}\right)+2\ln \left(\frac{\mu X+2{\mu }^2-\kappa x}{x\sqrt{{\kappa }^2-{\mu }^2}}\right),$$ ,

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

…

9: 4.45 Methods of Computation

… ►The inverses $\mathrm{arcsinh}$, $\mathrm{arccosh}$, and $\mathrm{arctanh}$ can be computed from the logarithmic forms given in §4.37(iv), with real arguments. …

10: 4.23 Inverse Trigonometric Functions

… ►

4.23.42 $${\mathrm{gd}}^{-1}\left(x\right)=\ln \tan \left(\frac{1}{2}x+\frac{1}{4}\pi \right)=\ln \left(\sec x+\tan x\right)=\mathrm{arcsinh}\left(\tan x\right)=\mathrm{arccsch}\left(\cot x\right)=\mathrm{arccosh}\left(\sec x\right)=\mathrm{arcsech}\left(\cos x\right)=\mathrm{arctanh}\left(\sin x\right)=\mathrm{arccoth}\left(\csc x\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $\cot z$: cotangent function, $\mathrm{arccsch}z$: inverse hyperbolic cosecant function, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{arccoth}z$: inverse hyperbolic cotangent function, $\mathrm{arcsech}z$: inverse hyperbolic secant function, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\ln z$: principal branch of logarithm function, $\sec z$: secant function, $\sin z$: sine function, $\tan z$: tangent function and $x$: real variable

Referenced by:

§19.9(ii), §4.23(viii), §4.26(ii)

Permalink:

http://dlmf.nist.gov/4.23.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(viii), §4.23 and Ch.4