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 \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}}

${\displaystyle {\begin{aligned}&\mathrm {kei} _{n}(x)=-\ln \left({\frac {x}{2}}\right)\mathrm {bei} _{n}(x)-{\frac {\pi }{4}}\mathrm {ber} _{n}(x)\\&-{\frac {1}{2}}\left({\frac {x}{2}}\right)^{-n}\sum _{k=0}^{n-1}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {(n-k-1)!}{k!}}\left({\frac {x^{2}}{4}}\right)^{k}\\&+{\frac {1}{2}}\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {\psi (k+1)+\psi (n+k+1)}{k!(n+k)!}}\left({\frac {x^{2}}{4}}\right)^{k}.\end{aligned}}}$

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

Semantic latex: \begin{align}&\Kelvinkei{n}@@{(x)} = - \ln(\frac{x}{2}) \Kelvinbei{n}@@{(x)} - \frac{\cpi}{4} \Kelvinber{n}@@{(x)} \\ &- \frac{1}{2}(\frac{x}{2})^{-n} \sum_{k=0}^{n-1} \sin [(\frac{3n}{4} + \frac{k}{2}) \cpi] \frac{(n-k-1)!}{k!}(\frac{x^2}{4})^k \\ &+ \frac{1}{2}(\frac{x}{2})^n \sum_{k \geq 0} \sin [(\frac{3n}{4} + \frac{k}{2}) \cpi] \frac{\digamma@{k + 1} + \digamma@{n + k + 1}}{k! (n+k)!}(\frac{x^2}{4})^k .\end{align}

Confidence: 0.66951655463217

Mathematica

Translation: KelvinKei[n, x] == - Log[Divide[x,2]]*KelvinBei[n, x]-Divide[Pi,4]*KelvinBer[n, x] -Divide[1,2]*(Divide[x,2])^(- n)* Sum[Sin[(Divide[3*n,4]+Divide[k,2])*Pi]*Divide[(n - k - 1)!,(k)!]*(Divide[(x)^(2),4])^(k), {k, 0, n - 1}, GenerateConditions->None] +Divide[1,2]*(Divide[x,2])^(n)* Sum[Sin[(Divide[3*n,4]+Divide[k,2])*Pi]*Divide[PolyGamma[k + 1]+ PolyGamma[n + k + 1],(k)!*(n + k)!]*(Divide[(x)^(2),4])^(k), {k, 0, Infinity}, GenerateConditions->None]

Information

Sub Equations

KelvinKei[n, x] = - Log[Divide[x,2]]*KelvinBei[n, x]-Divide[Pi,4]*KelvinBer[n, x]

Free variables

n

x

Symbol info

Kelvin Function KEI; Example: \Kelvinkei{\nu}@@{x}

Will be translated to: KelvinKei[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.61#E2 Mathematica: https://reference.wolfram.com/language/ref/KelvinKei.html

Kelvin Function BER; Example: \Kelvinber{\nu}@@{x}

Will be translated to: KelvinBer[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.61#E1 Mathematica: https://reference.wolfram.com/language/ref/KelvinBer.html

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

Digamma / Psi function; Example: \digamma@{z}

Will be translated to: PolyGamma[$0] Constraints: z not element of {0, -1, -2, ...} Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E2 Mathematica: https://reference.wolfram.com/language/ref/PolyGamma.html

Pi was translated to: Pi

Kelvin Function BEI; Example: \Kelvinbei{\nu}@@{x}

Will be translated to: KelvinBei[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.61#E1 Mathematica: https://reference.wolfram.com/language/ref/KelvinBei.html

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

Tests

Symbolic

Test expression: (KelvinKei[n, x])-(- Log[Divide[x,2]]*KelvinBei[n, x]-Divide[Pi,4]*KelvinBer[n, x])

ERROR:

{
    "result": "ERROR",
    "testTitle": "Simple",
    "testExpression": null,
    "resultExpression": null,
    "wasAborted": false,
    "conditionallySuccessful": false
}

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \Kelvinkei [\Kelvinkei]

Tests

Symbolic

Numeric

Maple

Translation: KelvinKei(n, x) = - ln((x)/(2))*KelvinBei(n, x)-(Pi)/(4)*KelvinBer(n, x) -(1)/(2)*((x)/(2))^(- n)* sum(sin(((3*n)/(4)+(k)/(2))*Pi)*(factorial(n - k - 1))/(factorial(k))*(((x)^(2))/(4))^(k), k = 0..n - 1) +(1)/(2)*((x)/(2))^(n)* sum(sin(((3*n)/(4)+(k)/(2))*Pi)*(Psi(k + 1)+ Psi(n + k + 1))/(factorial(k)*factorial(n + k))*(((x)^(2))/(4))^(k), k = 0..infinity)

Information

Sub Equations

KelvinKei(n, x) = - ln((x)/(2))*KelvinBei(n, x)-(Pi)/(4)*KelvinBer(n, x)

Free variables

n

x

Symbol info

Kelvin Function KEI; Example: \Kelvinkei{\nu}@@{x}

Will be translated to: KelvinKei($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.61#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=KelvinKei

Kelvin Function BER; Example: \Kelvinber{\nu}@@{x}

Will be translated to: KelvinBer($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.61#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=KelvinBer

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

Digamma / Psi function; Example: \digamma@{z}

Will be translated to: Psi($0) Constraints: z not element of {0, -1, -2, ...} Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Psi

Pi was translated to: Pi

Kelvin Function BEI; Example: \Kelvinbei{\nu}@@{x}

Will be translated to: KelvinBei($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.61#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=KelvinBei

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

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\psi (z)$

$x)$

$n$

$_{n}(x)$

$x$

${\begin{aligned}&\mathrm {kei} _{n}(x)=-\ln \left({\frac {x}{2}}\right)\mathrm {bei} _{n}(x)-{\frac {\pi }{4}}\mathrm {ber} _{n}(x)\\&-{\frac {1}{2}}\left({\frac {x}{2}}\right)^{-n}\sum _{k=0}^{n-1}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {(n-k-1)!}{k!}}\left({\frac {x^{2}}{4}}\right)^{k}\\&+{\frac {1}{2}}\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {\psi (k+1)+\psi (n+k+1)}{k!(n+k)!}}\left({\frac {x^{2}}{4}}\right)^{k}.\end{aligned}}$

Is part of

${\begin{aligned}&\mathrm {kei} _{n}(x)=-\ln \left({\frac {x}{2}}\right)\mathrm {bei} _{n}(x)-{\frac {\pi }{4}}\mathrm {ber} _{n}(x)\\&-{\frac {1}{2}}\left({\frac {x}{2}}\right)^{-n}\sum _{k=0}^{n-1}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {(n-k-1)!}{k!}}\left({\frac {x^{2}}{4}}\right)^{k}\\&+{\frac {1}{2}}\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {\psi (k+1)+\psi (n+k+1)}{k!(n+k)!}}\left({\frac {x^{2}}{4}}\right)^{k}.\end{aligned}}$

Complete translation information:

{
  "id" : "FORMULA_62dd8602607f0e5796a59a30eac17b51",
  "formula" : "\\begin{align}\n&\\mathrm{kei}_n(x) = - \\ln\\left(\\frac{x}{2}\\right) \\mathrm{bei}_n(x) - \\frac{\\pi}{4}\\mathrm{ber}_n(x) \\\\\n&-\\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  \\\\\n&+ \\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 .\n\\end{align}",
  "semanticFormula" : "\\begin{align}&\\Kelvinkei{n}@@{(x)} = - \\ln(\\frac{x}{2}) \\Kelvinbei{n}@@{(x)} - \\frac{\\cpi}{4} \\Kelvinber{n}@@{(x)} \\\\ &- \\frac{1}{2}(\\frac{x}{2})^{-n} \\sum_{k=0}^{n-1} \\sin [(\\frac{3n}{4} + \\frac{k}{2}) \\cpi] \\frac{(n-k-1)!}{k!}(\\frac{x^2}{4})^k \\\\ &+ \\frac{1}{2}(\\frac{x}{2})^n \\sum_{k \\geq 0} \\sin [(\\frac{3n}{4} + \\frac{k}{2}) \\cpi] \\frac{\\digamma@{k + 1} + \\digamma@{n + k + 1}}{k! (n+k)!}(\\frac{x^2}{4})^k .\\end{align}",
  "confidence" : 0.6695165546321705,
  "translations" : {
    "Mathematica" : {
      "translation" : "KelvinKei[n, x] == - Log[Divide[x,2]]*KelvinBei[n, x]-Divide[Pi,4]*KelvinBer[n, x] -Divide[1,2]*(Divide[x,2])^(- n)* Sum[Sin[(Divide[3*n,4]+Divide[k,2])*Pi]*Divide[(n - k - 1)!,(k)!]*(Divide[(x)^(2),4])^(k), {k, 0, n - 1}, GenerateConditions->None] +Divide[1,2]*(Divide[x,2])^(n)* Sum[Sin[(Divide[3*n,4]+Divide[k,2])*Pi]*Divide[PolyGamma[k + 1]+ PolyGamma[n + k + 1],(k)!*(n + k)!]*(Divide[(x)^(2),4])^(k), {k, 0, Infinity}, GenerateConditions->None]",
      "translationInformation" : {
        "subEquations" : [ "KelvinKei[n, x] = - Log[Divide[x,2]]*KelvinBei[n, x]-Divide[Pi,4]*KelvinBer[n, x]" ],
        "freeVariables" : [ "n", "x" ],
        "tokenTranslations" : {
          "\\Kelvinkei" : "Kelvin Function KEI; Example: \\Kelvinkei{\\nu}@@{x}\nWill be translated to: KelvinKei[$0, $1]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/10.61#E2\nMathematica:  https://reference.wolfram.com/language/ref/KelvinKei.html",
          "\\Kelvinber" : "Kelvin Function BER; Example: \\Kelvinber{\\nu}@@{x}\nWill be translated to: KelvinBer[$0, $1]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/10.61#E1\nMathematica:  https://reference.wolfram.com/language/ref/KelvinBer.html",
          "\\sin" : "Sine; Example: \\sin@@{z}\nWill be translated to: Sin[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/4.14#E1\nMathematica:  https://reference.wolfram.com/language/ref/Sin.html",
          "\\digamma" : "Digamma / Psi function; Example: \\digamma@{z}\nWill be translated to: PolyGamma[$0]\nConstraints: z not element of {0, -1, -2, ...}\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/5.2#E2\nMathematica:  https://reference.wolfram.com/language/ref/PolyGamma.html",
          "\\cpi" : "Pi was translated to: Pi",
          "\\Kelvinbei" : "Kelvin Function BEI; Example: \\Kelvinbei{\\nu}@@{x}\nWill be translated to: KelvinBei[$0, $1]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/10.61#E1\nMathematica:  https://reference.wolfram.com/language/ref/KelvinBei.html",
          "\\ln" : "Natural logarithm; Example: \\ln@@{z}\nWill be translated to: Log[$0]\nConstraints: z != 0\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/4.2#E2\nMathematica:  https://reference.wolfram.com/language/ref/Log.html"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "ERROR",
        "numberOfTests" : 1,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 1,
        "crashed" : false,
        "testCalculationsGroup" : [ {
          "lhs" : "KelvinKei[n, x]",
          "rhs" : "- Log[Divide[x,2]]*KelvinBei[n, x]-Divide[Pi,4]*KelvinBer[n, x]",
          "testExpression" : "(KelvinKei[n, x])-(- Log[Divide[x,2]]*KelvinBei[n, x]-Divide[Pi,4]*KelvinBer[n, x])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\Kelvinkei [\\Kelvinkei]"
        }
      }
    },
    "Maple" : {
      "translation" : "KelvinKei(n, x) = - ln((x)/(2))*KelvinBei(n, x)-(Pi)/(4)*KelvinBer(n, x) -(1)/(2)*((x)/(2))^(- n)* sum(sin(((3*n)/(4)+(k)/(2))*Pi)*(factorial(n - k - 1))/(factorial(k))*(((x)^(2))/(4))^(k), k = 0..n - 1) +(1)/(2)*((x)/(2))^(n)* sum(sin(((3*n)/(4)+(k)/(2))*Pi)*(Psi(k + 1)+ Psi(n + k + 1))/(factorial(k)*factorial(n + k))*(((x)^(2))/(4))^(k), k = 0..infinity)",
      "translationInformation" : {
        "subEquations" : [ "KelvinKei(n, x) = - ln((x)/(2))*KelvinBei(n, x)-(Pi)/(4)*KelvinBer(n, x)" ],
        "freeVariables" : [ "n", "x" ],
        "tokenTranslations" : {
          "\\Kelvinkei" : "Kelvin Function KEI; Example: \\Kelvinkei{\\nu}@@{x}\nWill be translated to: KelvinKei($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/10.61#E2\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=KelvinKei",
          "\\Kelvinber" : "Kelvin Function BER; Example: \\Kelvinber{\\nu}@@{x}\nWill be translated to: KelvinBer($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/10.61#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=KelvinBer",
          "\\sin" : "Sine; Example: \\sin@@{z}\nWill be translated to: sin($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/4.14#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin",
          "\\digamma" : "Digamma / Psi function; Example: \\digamma@{z}\nWill be translated to: Psi($0)\nConstraints: z not element of {0, -1, -2, ...}\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/5.2#E2\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Psi",
          "\\cpi" : "Pi was translated to: Pi",
          "\\Kelvinbei" : "Kelvin Function BEI; Example: \\Kelvinbei{\\nu}@@{x}\nWill be translated to: KelvinBei($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/10.61#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=KelvinBei",
          "\\ln" : "Natural logarithm; Example: \\ln@@{z}\nWill be translated to: ln($0)\nConstraints: z != 0\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/4.2#E2\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=ln"
        }
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "\\psi(z)", "x)", "n", "_{n}(x)", "x", "\\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}" ],
  "isPartOf" : [ "\\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}" ],
  "definiens" : [ ]
}

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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