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Display information for equation id:math.223294.14 on revision:223294

* Page found: Box–Muller transform (eq math.223294.14)

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Hash: bc16f5b0df2879fbfdd65f8c306eee09

TeX (original user input):

z_1 = \sqrt{-2 \ln U_1} \sin(2 \pi U_2) = \sqrt{-2 \ln s}\left( \frac{v}{\sqrt{s}}\right) = v \cdot \sqrt{\frac{-2 \ln s}{s}}.

TeX (checked):

z_{1}={\sqrt {-2\ln U_{1}}}\sin(2\pi U_{2})={\sqrt {-2\ln s}}\left({\frac {v}{\sqrt {s}}}\right)=v\cdot {\sqrt {\frac {-2\ln s}{s}}}.

LaTeXML (experimental; uses MathML) rendering

MathML (12.409 KB / 1.808 KB) :

z 1 = - 2 ln U 1 sin ( 2 π U 2 ) = - 2 ln s ( v s ) = v - 2 ln s s . subscript z 1 2 subscript U 1 2 π subscript U 2 2 s v s normal-⋅ v 2 s s {\displaystyle z_{1}={\sqrt{-2\ln U_{1}}}\sin(2\pi U_{2})={\sqrt{-2\ln s}}% \left({\frac{v}{{\sqrt{s}}}}\right)=v\cdot{\sqrt{{\frac{-2\ln s}{s}}}}.}
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    <annotation encoding="application/x-tex" id="p1.1.m1.1c">{\displaystyle z_{1}={\sqrt{-2\ln U_{1}}}\sin(2\pi U_{2})={\sqrt{-2\ln s}}%
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z 1 equals StartRoot minus 2 times ln upper U 1 EndRoot times sine left-parenthesis 2 times pi times upper U 2 right-parenthesis equals StartRoot minus 2 times ln s EndRoot times left-parenthesis StartFraction v Over StartRoot s EndRoot EndFraction right-parenthesis equals v dot StartRoot StartFraction minus 2 times ln s Over s EndFraction EndRoot period

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Translations to Computer Algebra Systems

Translation to Maple

In Maple: z[1]=sqrt(- 2*ln(U)[1])*sin(2*pi*U[2])=sqrt(- 2*ln(s))*((v)/(sqrt(s)))= v *sqrt((- 2*ln(s))/(s))

Information about the conversion process:

\sin: Sine; Example: \sin@@{z}

Will be translated to: sin($0)

Relevant links to definitions:

DLMF: http://dlmf.nist.gov/4.14#E1

Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin


\ln: Natural logarithm; Example: \ln@@{z}

Will be translated to: ln($0)

Constraints: z != 0

Branch Cuts: (-\infty, 0]

Relevant links to definitions:

DLMF: http://dlmf.nist.gov/4.2#E2

Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=ln


\cdot: was translated to: *

\pi: Could be the ratio of a circle's circumference to its diameter == Archimedes' constant.

But it is also a Greek letter. Be aware, that this program translated the letter as a normal Greek letter and not as a constant!

Use the DLMF-Macro \cpi to translate \pi as a constant.



Translation to Mathematica

In Mathematica: Subscript[z, 1]=Sqrt[- 2*Subscript[Log[U], 1]]*Sin[2*\[Pi]*Subscript[U, 2]]=Sqrt[- 2*Log[s]]*(Divide[v,Sqrt[s]])= v *Sqrt[Divide[- 2*Log[s],s]]

Information about the conversion process:

\sin: Sine; Example: \sin@@{z}

Will be translated to: Sin[$0]

Relevant links to definitions:

DLMF: http://dlmf.nist.gov/4.14#E1

Mathematica: https://reference.wolfram.com/language/ref/Sin.html


\ln: Natural logarithm; Example: \ln@@{z}

Will be translated to: Log[$0]

Constraints: z != 0

Branch Cuts: (-\infty, 0]

Relevant links to definitions:

DLMF: http://dlmf.nist.gov/4.2#E2

Mathematica: https://reference.wolfram.com/language/ref/Log.html


\cdot: was translated to: *

\pi: Could be the ratio of a circle's circumference to its diameter == Archimedes' constant.

But it is also a Greek letter. Be aware, that this program translated the letter as a normal Greek letter and not as a constant!

Use the DLMF-Macro \cpi to translate \pi as a constant.



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