Real elementary functions

• Root, exponential, logarithmic and power functions
  • Auxiliary function \(\mathrm{sqrt1pmx}(x) = \sqrt{1+x^2}-x\)
    • math53.sqrt1pmx()
  • Bring radical
    • math53.bring()
  • Auxiliary function \(\mathrm{expmx2h}(x) = \exp(-x^2 / 2)\)
    • math53.expmx2h()
  • Relative error exponential \(\mathrm{exprel}(x) = (\exp(x) - 1)/x\)
    • math53.exprel()
  • Auxiliary function \(\mathrm{expx2}(x) = \exp(x \cdot |x|)\)
    • math53.expx2()
  • Auxiliary function \(\mathrm{logistic}(x) = 1/(1+\exp(-x))\)
    • math53.logistic()
  • Einstein functions
    • math53.einstein()
  • Auxiliary function \(\mathrm{log1mexp}(x) = \log(1-\exp(-|x|))\)
    • math53.log1mexp()
  • Auxiliary function \(\mathrm{log1pexp}(x) = \log(1+\exp(x))\)
    • math53.log1pexp()
  • Auxiliary function \(\mathrm{log1pmx}(x) = \log(1+x)-x\)
    • math53.log1pmx()
  • Auxiliary function \(\mathrm{logaddexp}(x, y) = \log[\exp(x) + \exp(y)]\)
    • math53.logaddexp()
  • Auxiliary function \(\mathrm{logsubexp}(x, y) = \log[\exp(x) - \exp(y)]\)
    • math53.logsubexp()
  • Auxiliary function \(\mathrm{logit}(x) = \log(x/(1.0-x))\)
    • math53.logit()
  • Wright \(\omega\) function
    • math53.wright_omega()
  • Auxiliary function \(\mathrm{compound}(x,n) = (1+x)^n\)
    • ctx.compound()
  • Auxiliary function \(\mathrm{comprel}(x,n) = (1+x)^n - 1\)
    • ctx.comprel()
  • Auxiliary function \(\mathrm{hypot3}(x,y,z) = \sqrt{x^2 + y^2 + z^2}\)
    • math53.hypot3()
  • Fibonacci polynomials, \(\mathrm{fibpoly}(n, x)\)
    • math53.fibpoly()
  • Lucas polynomials, \(\mathrm{lucaspoly}(n, x)\)
    • math53.lucaspoly()
  • Fibonacci function, \(F_{\nu}(x)\), of real index \(\nu\)
    • math53.fibfun()
• Trigonometric functions
  • Sine, \(x\) in degrees, \(\mathrm{sind}(x)\)
    • math53.sind()
  • Inverse sine, input in degrees, \(\mathrm{asind}(x)\)
    • math53.asind()
  • Cosine, \(x\) in degrees, \(\mathrm{cosd}(x)\)
    • math53.cosd()
  • Inverse cosine, input in degrees, \(\mathrm{acosd}(x)\)
    • math53.acosd()
  • Tangent, with \(x\) in degrees, \(\mathrm{tand}(x)\)
    • math53.tand()
  • Inverse tangent, input in degrees, \(\mathrm{atand}(x)\)
    • math53.atand()
  • Cotangent, with \(x\) in degrees, \(\mathrm{cotd}(x)\)
    • math53.cotd()
  • Inverse cotangent, input in degrees, \(\mathrm{acotd}(x)\)
    • math53.acotd()
  • Auxiliary function, \(\mathrm{acos}(1-x)\)
    • math53.acos1m()
  • Haversine function \(\mathrm{hav}(x) = (1 - \cos(x))/2\)
    • math53.hav()
  • Inverse haversine function \(\mathrm{archav}(x) = \mathrm{acos}(1-2x)\)
    • math53.archav()
  • Versine function \(\mathrm{vers}(x) = 1 - \cos(x)\)
    • math53.vers()
  • Coversine, \(\mathrm{covers}(x) = 1 - \sin(x)\)
    • math53.covers()
  • Versint function \(\mathrm{versint}(x) = x - \sin(x)\)
    • math53.versint()
  • Integral of cos powers, \(\mathrm{cosint}(n,x)\)
    • math53.cosint()
  • Integral of sin powers, \(\mathrm{sinint}(n,x)\)
    • math53.sinint()
• Hyperbolic functions
  • Cardinal hyperbolic sine, \(\mathrm{sinhc}(x) = \sinh(x)/x\)
    • math53.sinhc()
  • Auxiliary function, \(\mathrm{sinhmx}(x) = \sinh(x)-x\)
    • math53.sinhmx()
  • Auxiliary function, \(\mathrm{coshm1}(x) = \cosh(x)-1\)
    • math53.coshm1()
  • Auxiliary function \(\mathrm{acosh}(1+x)\)
    • math53.acosh1p()
  • Auxiliary function \(\log(\cosh(x))\)
    • math53.logcosh()
  • Auxiliary function \(\log(\sinh(x))\)
    • math53.logsinh()
  • Gudermannian function \(\mathrm{gd}(x) = \mathrm{asin}(\mathrm{tanh}(x))\)
    • math53.gd()
  • Inverse Gudermannian function \(\mathrm{arcgd}(x) = \mathrm{atanh}(\sin(x))\)
    • math53.arcgd()
  • Langevin function, \(L(x)\)
    • math53.langevin_l()
  • Inverse Langevin function, \(L^{-1}(x)\)
    • math53.langevin_l_inv()
  • Solutions of Kepler’s equation, \(\mathrm{kepler}(M,e)\)
    • math53.kepler()
• Gamma functions
  • Sign of the gamma function
    • math53.signgamma()
  • Logarithm and sign of the gamma function
    • math53.lgamma_s()
  • Logarithm of \(\Gamma(1 + x)\)
    • math53.lgamma1p()
  • Logarithm of factorials: \(\log(x!)\)
    • math53.logfactorial()
  • Logarithm of the binomial coefficient
    • math53.logbinomial()
  • Log-Beta function
    • math53.logbeta()
  • Inverse of the gamma function, \(\Gamma^{-1}(y)\)
    • math53.gamma_inv()
  • Inverse of the logarithm of the gamma function, \(\log\Gamma^{-1}(y)\)
    • math53.lgamma_inv()
  • Temme’s regulated gamma function, \(\Gamma^{*}(x)\)
    • math53.gammastar()
  • Relative Pochhammer symbol, \(((a)_x - 1)/x\)
    • math53.poch1()
  • Catalan function \(C(x)\)
    • math53.catalan_c()