Addditional elementary functions (real arguments, double precision)

General concepts

See also: https://en.wikipedia.org/wiki/Power_of_a_test

See also: https://en.wikipedia.org/wiki/Sample_size_determination

• Additional root, exponential, logarithmic and power functions
  • Auxiliary function \(\mathrm{sqrt1pmx}(x) = \sqrt{1+x^2}-x\)
    • math53.sqrt1pmx()
  • Cube root, \(\mathrm{cuberoot}(x) = \sqrt[3]{x} = y\), with \(\mathrm{arg}(y)\) closest to \(\mathrm{arg}(x)\)
    • math53.cuberoot()
  • Nth root, \(\mathrm{surd}(x, n) = \sqrt[n]{x} = y\), with \(\mathrm{arg}(y)\) closest to \(\mathrm{arg}(x)\)
    • mathc53.surd()
  • Bring radical
    • math53.bring()
  • Auxiliary function \(\mathrm{expmx2h}(x) = \exp(-x^2 / 2)\)
    • math53.expmx2h()
  • Relative error exponential \(\mathrm{exprel}(x) = (\exp(x) - 1)/x\)
    • ctx.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 \(\mathrm{hypot3}(x,y,z) = \sqrt{x^2 + y^2 + z^2}\)
    • math53.hypot3()
  • Fibonacci polynomials, \(\mathrm{fibpoly}(n, x)\)
    • ctx.fibpoly()
  • Lucas polynomials, \(\mathrm{lucaspoly}(n, x)\)
    • ctx.lucaspoly()
• Additional Trigonometric functions (real arguments only)
  • Sine, \(x\) in degrees, \(\mathrm{sind}(x)\)
    • math53.sind()
  • Inverse sine, input in degrees, \(\mathrm{asind}(x)\)
    • ctx.asind()
  • Cosine, \(x\) in degrees, \(\mathrm{cosd}(x)\)
    • math53.cosd()
  • Inverse cosine, input in degrees, \(\mathrm{acosd}(x)\)
    • ctx.acosd()
  • Tangent, with \(x\) in degrees, \(\mathrm{tand}(x)\)
    • math53.tand()
  • Inverse tangent, input in degrees, \(\mathrm{atand}(x)\)
    • ctx.atand()
  • Cotangent, with \(x\) in degrees, \(\mathrm{cotd}(x)\)
    • math53.cotd()
  • Inverse cotangent, input in degrees, \(\mathrm{acotd}(x)\)
    • ctx.acotd()
  • Continuous inverse cotangent, \(\mathrm{acotc}(x)\)
    • ctx.acotc()
  • Coversine, \(\mathrm{covers}(x) = 1 - \sin(x)\)
    • math53.covers()
  • Versint function \(\mathrm{versint}(x) = x - \sin(x)\)
    • math53.versint()
  • Versine function \(\mathrm{vers}(x) = 1 - \cos(x)\)
    • math53.versine()
  • Haversine function \(\mathrm{hav}(x) = (1 - \cos(x))/2\)
    • math53.haversine()
  • Integral of cos powers, \(\mathrm{cosint}(n,x)\)
    • math53.cosint()
  • Integral of sin powers, \(\mathrm{sinint}(n,x)\)
    • math53.sinint()
  • Solutions of Kepler’s equation, \(\mathrm{kepler}(M,e)\)
    • math53.kepler()
  • Fibonacci function, \(F_{\nu}(x)\), of real index \(\nu\)
    • math53.fibfun()
  • 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()
  • Langevin function, \(L(x)\)
    • math53.langevin_l()
  • Auxiliary function \(\log(\cosh(x))\)
    • math53.logcosh()
  • Auxiliary function \(\log(\sinh(x))\)
    • math53.logsinh()
  • Auxiliary function, \(\mathrm{acos}(1-x)\)
    • math53.acos1m()
  • Gudermannian function \(\mathrm{gd}(x) = \mathrm{asin}(\mathrm{tanh}(x))\)
    • math53.gd()
  • Inverse haversine function \(\mathrm{archav}(x) = \mathrm{acos}(1-2x)\)
    • math53.archav()
  • Auxiliary function \(\mathrm{acosh}(1+x)\)
    • math53.acosh1p()
  • Inverse Gudermannian function \(\mathrm{arcgd}(x) = \mathrm{atanh}(\sin(x))\)
    • math53.arcgd()
  • Inverse Langevin function, \(L^{-1}(x)\)
    • math53.langevin_l_inv()
• Additional real error functions (real arguments only)
  • Exponentially scaled complementary error function, \(\mathrm{erfcx}(x)\)
    • math53.real_erfcx()
  • Imaginary error function, \(\mathrm{erfi}(x)\)
    • ctx.real_erfi()
  • Difference of error functions, \(\mathrm{erfh}(x,h) = \mathrm{erf}(x+h)-\mathrm{erf}(x-h)\)
    • math53.real_erfh()
  • Difference of error functions, \(\mathrm{erf2}(x_1,x_2) = \mathrm{erf}(x_2)-\mathrm{erf}(x_1)\)
    • math53.real_erf2()
  • Probability function \(Q(x) = \Phi(-x)\)
    • math53.real_erfq()
  • Standard normal quantile function \(\Phi^{-1}(q)\)
    • ctx.ndisx()
  • Inverse of the exponentially scaled complementary error function, \(\mathrm{erfcx}^{-1}(x)\)
    • math53.real_erfcx_inv()
  • Inverse of the imaginary error function, \(\mathrm{erfi}^{-1}(x)\)
    • math53.real_erfi_inv()
• Additional real gamma functions (real arguments only)
  • Reciprocal Gamma function, \(1/\Gamma(x)\)
    • ctx.real_rgamma()
  • Logarithm of \(\Gamma(1 + x)\)
    • math53.real_lgamma1p()
  • Sign of the gamma function
    • math53.real_signgamma()
  • Logarithm and sign of the gamma function
    • math53.real_lgamma_s()
  • Temme’s regulated gamma function, \(\Gamma^{*}(x)\)
    • math53.real_gammastar()
  • Logarithm of factorials: \(\log(x!)\)
    • math53.real_logfactorial()
  • Catalan function \(C(x)\)
    • ctx.catalan_c()
  • Logarithm of the binomial coefficient
    • math53.real_logbinomial()
  • Inverse of the gamma function, \(\Gamma^{-1}(y)\)
    • math53.real_gamma_inv()
  • Inverse of the logarithm of the gamma function, \(\log\Gamma^{-1}(y)\)
    • math53.real_lgamma_inv()
  • Relative Pochhammer symbol, \(((a)_x - 1)/x\)
    • math53.real_poch1()
  • Beta function, \(B(a,b)\)
    • ctx.real_beta()
  • Log-Beta function
    • ctx.real_logbeta()
• Additional real incomplete gamma functions (real arguments only)
  • Tricomi’s entire incomplete gamma function: \(\gamma^*(a,x)\)
    • ctx.real_gamma_tricomi()
  • Truncated exponential function, \(e_n(x)\)
    • math53.expn()
  • Relative exponential, \(\mathrm{exprel}_n(x)\)
    • math53.expreln()