LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle f_1(x)}

${\displaystyle f_{1}(x)}$

... is translated to the CAS output ...

Semantic latex: f_1(x)

Confidence: 0

Mathematica

Translation: Subscript[f, 1][x]

Information

Sub Equations

Subscript[f, 1][x]

Free variables

x

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

SymPy

Translation: Symbol('{f}_{1}')(x)

Information

Sub Equations

Symbol('{f}_{1}')(x)

Free variables

x

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

Maple

Translation: f[1](x)

Information

Sub Equations

f[1](x)

Free variables

x

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$x)$

$x$

Is part of

$\mathrm {bei} (x)\sim {\frac {e^{\frac {x}{\sqrt {2}}}}{\sqrt {2\pi x}}}[f_{1}(x)\sin \alpha -g_{1}(x)\cos \alpha ]-{\frac {\mathrm {ker} (x)}{\pi }}$

$\mathrm {ber} (x)\sim {\frac {e^{\frac {x}{\sqrt {2}}}}{\sqrt {2\pi x}}}\left(f_{1}(x)\cos \alpha +g_{1}(x)\sin \alpha \right)-{\frac {\mathrm {kei} (x)}{\pi }}$

$f_{1}(x)=1+\sum _{k\geq 1}{\frac {\cos(k\pi /4)}{k!(8x)^{k}}}\prod _{l=1}^{k}(2l-1)^{2}$

Description

asymptotic series

series expansion

special case

ber

bei

Complete translation information:

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  }, {
    "definition" : "ber",
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  }, {
    "definition" : "bei",
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  } ]
}

Specify your own input

Formula input
Wikitext context	In applied mathematics, the '''Kelvin functions''' ber<sub>''ν''</sub>(''x'') and bei<sub>''ν''</sub>(''x'') are the [[real part\|real]] and [[imaginary part]]s, respectively, of :<math>J_\nu \left (x e^{\frac{3 \pi i}{4}} \right ),\,</math><!-- Do not delete "\," it improves display of formula on certain browsers. ---> where ''x'' is real, and {{math\|''J<sub>ν</sub>''(''z'')}}, is the ''ν''<sup>th</sup> order [[Bessel function]] of the first kind. Similarly, the functions ker<sub>ν</sub>(''x'') and kei<sub>ν</sub>(''x'') are the real and imaginary parts, respectively, of :<math>K_\nu \left (x e^{\frac{\pi i}{4}} \right ),\,</math> <!-- Do not delete "\," it improves display of formula on certain browsers. ---> where {{math\|''K<sub>ν</sub>''(''z'')}} is the ''ν''<sup>th</sup> order [[Bessel function#Modified Bessel functions\|modified Bessel function]] of the second kind. These functions are named after [[William Thomson, 1st Baron Kelvin]]. While the Kelvin functions are defined as the real and imaginary parts of Bessel functions with ''x'' taken to be real, the functions can be analytically continued for complex arguments {{math\|''xe''<sup>''iφ''</sup>, 0 ≤ ''φ'' < 2''π''.}} With the exception of ber<sub>''n''</sub>(''x'') and bei<sub>''n''</sub>(''x'') for integral ''n'', the Kelvin functions have a [[branch point]] at ''x'' = 0. Below, {{math\|Γ(''z'')}} is the [[gamma function]] and {{math\|''ψ''(''z'')}} is the [[digamma function]]. == ber(''x'') == [[Image:Mplwp Kelvin-ber.svg\|thumb\|right\|ber(''x'') for ''x'' between 0 and 20.]] [[Image:Mplwp Kelvin-ber-norm.svg\|thumb\|right\|<math>\mathrm{ber}(x) / e^{x/\sqrt{2}}</math> for ''x'' between 0 and 50.]] For integers ''n'', ber<sub>''n''</sub>(''x'') has the series expansion :<math>\mathrm{ber}_n(x) = \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \frac{\cos\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right]}{k! \Gamma(n + k + 1)} \left(\frac{x^2}{4}\right)^k ,</math> where {{math\|Γ(''z'')}} is the [[gamma function]]. The special case ber<sub>0</sub>(''x''), commonly denoted as just ber(''x''), has the series expansion :<math>\mathrm{ber}(x) = 1 + \sum_{k \geq 1} \frac{(-1)^k}{[(2k)!]^2} \left(\frac{x}{2} \right )^{4k}</math> and [[asymptotic series]] :<math>\mathrm{ber}(x) \sim \frac{e^{\frac{x}{\sqrt{2}}}}{\sqrt{2 \pi x}} \left (f_1(x) \cos \alpha + g_1(x) \sin \alpha \right ) - \frac{\mathrm{kei}(x)}{\pi}</math>, where :<math>\alpha = \frac{x}{\sqrt{2}} - \frac{\pi}{8},</math> :<math>f_1(x) = 1 + \sum_{k \geq 1} \frac{\cos(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2</math> :<math>g_1(x) = \sum_{k \geq 1} \frac{\sin(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2 .</math> == bei(''x'') == [[Image:Mplwp Kelvin-bei.svg\|thumb\|right\|bei(''x'') for ''x'' between 0 and 20.]] [[Image:Mplwp Kelvin-bei-norm.svg\|thumb\|right\|<math>\mathrm{bei}(x) / e^{x/\sqrt{2}}</math> for ''x'' between 0 and 50.]] For integers ''n'', bei<sub>''n''</sub>(''x'') has the series expansion :<math>\mathrm{bei}_n(x) = \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \frac{\sin\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right]}{k! \Gamma(n + k + 1)} \left(\frac{x^2}{4}\right)^k .</math> The special case bei<sub>0</sub>(''x''), commonly denoted as just bei(''x''), has the series expansion :<math>\mathrm{bei}(x) = \sum_{k \geq 0} \frac{(-1)^k }{[(2k+1)!]^2} \left(\frac{x}{2} \right )^{4k+2}</math> and asymptotic series :<math>\mathrm{bei}(x) \sim \frac{e^{\frac{x}{\sqrt{2}}}}{\sqrt{2 \pi x}} [f_1(x) \sin \alpha - g_1(x) \cos \alpha] - \frac{\mathrm{ker}(x)}{\pi},</math> where α, <math>f_1(x)</math>, and <math>g_1(x)</math> are defined as for ber(''x''). {{clear}} == ker(''x'') == [[Image:Mplwp Kelvin-ker.svg\|thumb\|right\|ker(''x'') for ''x'' between 0 and 14.]] [[Image:Mplwp Kelvin-ker-norm.svg\|thumb\|right\|<math>\mathrm{ker}(x) e^{x/\sqrt{2}}</math> for ''x'' between 0 and 50.]] For integers ''n'', ker<sub>''n''</sub>(''x'') has the (complicated) series expansion :<math>\begin{align} &\mathrm{ker}_n(x) = - \ln\left(\frac{x}{2}\right) \mathrm{ber}_n(x) + \frac{\pi}{4}\mathrm{bei}_n(x) \\ &+ \frac{1}{2} \left(\frac{x}{2}\right)^{-n} \sum_{k=0}^{n-1} \cos\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{(n-k-1)!}{k!} \left(\frac{x^2}{4}\right)^k \\ &+ \frac{1}{2} \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \cos\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{\psi(k+1) + \psi(n + k + 1)}{k! (n+k)!} \left(\frac{x^2}{4}\right)^k . \end{align}</math> The special case ker<sub>0</sub>(''x''), commonly denoted as just ker(''x''), has the series expansion :<math>\mathrm{ker}(x) = -\ln\left(\frac{x}{2}\right) \mathrm{ber}(x) + \frac{\pi}{4}\mathrm{bei}(x) + \sum_{k \geq 0} (-1)^k \frac{\psi(2k + 1)}{[(2k)!]^2} \left(\frac{x^2}{4}\right)^{2k}</math> and the asymptotic series :<math>\mathrm{ker}(x) \sim \sqrt{\frac{\pi}{2x}} e^{-\frac{x}{\sqrt{2}}} [f_2(x) \cos \beta + g_2(x) \sin \beta],</math> where :<math>\beta = \frac{x}{\sqrt{2}} + \frac{\pi}{8},</math> :<math>f_2(x) = 1 + \sum_{k \geq 1} (-1)^k \frac{\cos(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2</math> :<math>g_2(x) = \sum_{k \geq 1} (-1)^k \frac{\sin(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2.</math> {{clear}} == kei(''x'') == [[Image:Mplwp Kelvin-kei.svg\|thumb\|right\|kei(''x'') for ''x'' between 0 and 14.]] [[Image:Mplwp Kelvin-kei-norm.svg\|thumb\|right\|<math>\mathrm{kei}(x) e^{x/\sqrt{2}}</math> for ''x'' between 0 and 50.]] For integer ''n'', kei<sub>''n''</sub>(''x'') has the series expansion :<math>\begin{align} &\mathrm{kei}_n(x) = - \ln\left(\frac{x}{2}\right) \mathrm{bei}_n(x) - \frac{\pi}{4}\mathrm{ber}_n(x) \\ &-\frac{1}{2} \left(\frac{x}{2}\right)^{-n} \sum_{k=0}^{n-1} \sin\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{(n-k-1)!}{k!} \left(\frac{x^2}{4}\right)^k \\ &+ \frac{1}{2} \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \sin\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{\psi(k+1) + \psi(n + k + 1)}{k! (n+k)!} \left(\frac{x^2}{4}\right)^k . \end{align}</math> The special case kei<sub>0</sub>(''x''), commonly denoted as just kei(''x''), has the series expansion :<math>\mathrm{kei}(x) = -\ln\left(\frac{x}{2}\right) \mathrm{bei}(x) - \frac{\pi}{4}\mathrm{ber}(x) + \sum_{k \geq 0} (-1)^k \frac{\psi(2k + 2)}{[(2k+1)!]^2} \left(\frac{x^2}{4}\right)^{2k+1}</math> and the asymptotic series :<math>\mathrm{kei}(x) \sim -\sqrt{\frac{\pi}{2x}} e^{-\frac{x}{\sqrt{2}}} [f_2(x) \sin \beta + g_2(x) \cos \beta],</math> where ''β'', ''f''<sub>2</sub>(''x''), and ''g''<sub>2</sub>(''x'') are defined as for ker(''x''). {{clear}} == See also == * [[Bessel function]] == References == * {{Abramowitz_Stegun_ref\|9\|379}} {{dlmf\|first=F. W. J. \|last=Olver\|first2=L. C. \|last2=Maximon\|id=10\|title=Bessel functions}} == External links == Weisstein, Eric W. "Kelvin Functions." From MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/KelvinFunctions.html] * GPL-licensed C/C++ source code for calculating Kelvin functions at codecogs.com: [https://web.archive.org/web/20070407195618/http://www.codecogs.com/d-ox/maths/special/bessel/kelvin.php] [[Category:Special hypergeometric functions]] [[Category:William Thomson, 1st Baron Kelvin\|Functions]]
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LaTeX to CAS translator

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