Elementary scalar functions (real and complex)

The functions in this chapter comprise the usual elementary functions, as included in the C++ standard library, and a few specializations.

The source code for the C# tests in Visual Studio is divided by real and complex modules:

The source code for the C# tests in Visual Studio using real modules can be found online in the XlCalcNet repository or in the corresponding local XlCalcNet folder in the file B03a_RealElementaryFunctions.cs,

The source code for the C# tests in Visual Studio using complex modules can be found online in the XlCalcNet repository or in the corresponding local XlCalcNet folder in the file B03b_CplxElementaryFunctions.cs,

Elementary scalar functions

• Complex components
  • Absolute value of a real or complex number
    • ctx.abs()
    • ctx.fabs()
  • Sign of a real or complex number
    • ctx.sign()
  • Real part of a real or complex number
    • ctx.real()
  • Imaginary part of a real or complex number
    • ctx.imag()
  • Phase (or argument) of a real or complex number
    • ctx.phase()
  • Conjugate of a real or complex number
    • ctx.conj()
  • Polar representation of a real or complex number
    • ctx.polar()
  • Rectangular coordinates calculated from the polar representation of a real or complex number
    • ctx.rect()
• Roots and quadratic, cubic, and quartic equations
  • Square root, \(\mathrm{sqrt}(x) = \sqrt{x}\)
    • ctx.sqrt()
  • Reciprocal square root, \(\mathrm{rsqrt}(x) = 1/\sqrt{x}\)
    • ctx.rsqrt()
  • Auxiliary function \(\mathrm{sqrt1pm1}(x) = \sqrt{1+x}-1\)
    • ctx.sqrt1pm1()
  • Cube root, \(\mathrm{cbrt}(x) = \sqrt[3]{x}\)
    • ctx.cbrt()
  • Nth root, \(\mathrm{nroot}(x, n) = \sqrt[n]{x}\)
    • ctx.nroot()
  • Unit root, \(\mathrm{unitroot}(n)\)
    • ctx.unitroot()
• Exponential and related functions
  • Exponential function \(\exp(x) = e^x\)
    • ctx.exp()
  • Auxiliary function \(\mathrm{expj}(x) = e^{ix}\)
    • ctx.expj()
  • Auxiliary function \(\mathrm{expjpi}(x) = e^{i \pi x} = (-1)^x\)
    • ctx.expjpi()
  • Exponential function with base \(10\), \(\mathrm{exp10}(x) = 10^z\)
    • ctx.exp10()
  • Exponential function with base \(2\), \(\mathrm{exp2}(x) = 2^x\)
    • ctx.exp2()
  • Auxiliary function \(\mathrm{expm1}(x) = e^x-1\)
    • ctx.expm1()
  • Auxiliary function \(\mathrm{exp10m1}(x) = 10^x - 1\)
    • ctx.exp10m1()
  • Auxiliary function \(\mathrm{exp2m1}(x) = 2^x - 1\)
    • ctx.exp2m1()
• Logarithms and related functions
  • Natural logarithm \(\log(x)\)
    • ctx.log()
  • Logarithm with base \(10\), \(\mathrm{log10}(x) = \log_{10}(x)\)
    • ctx.log10()
  • Logarithm with base \(2\), \(\mathrm{log2}(x) = \log_{2}(x)\)
    • ctx.log2()
  • Logarithm with base \(b\), \(\mathrm{logbase}(x, b) = \log_{b}(x)\)
    • ctx.logbase()
  • Auxiliary function \(\mathrm{log1p}(x) = \log(x+1)\)
    • ctx.log1p()
  • Auxiliary function \(\mathrm{log10p1}(x) = \log_{10}(1+x)\)
    • ctx.log10p1()
  • Auxiliary function \(\mathrm{log2p1}(x) = \log_2(1+x)\)
    • ctx.log2p1()
• Power functions
  • Square, \(x^2\)
    • ctx.sqr()
  • Cube, \(x^3\)
    • ctx.cube()
  • Auxiliary function \(\mathrm{powi}(x,n) = x^n\)
    • ctx.powi()
  • Auxiliary function \(\mathrm{compound}(x,n) = (1+x)^n\)
    • math53.compound()
  • Auxiliary function \(\mathrm{comprel}(x,n) = (1+x)^n - 1\)
    • math53.comprel()
  • Auxiliary function \(\mathrm{hypot}(x,y) = \sqrt{x^2 + y^2}\)
    • ctx.hypot()
  • Power function, \(\mathrm{pow}(x, y) = x^y\)
    • ctx.pow()
  • Auxiliary function, \(\mathrm{powm1}(x, y) = x^y-1\)
    • ctx.powm1()
  • Auxiliary function, \(\mathrm{pow1p}(x, y) = (1+x)^y\)
    • ctx.pow1p()
  • Auxiliary function, \(\mathrm{pow1pm1}(x, y) = (1+x)^y - 1\)
    • ctx.pow1pm1()
• Trigonometric functions, in radians
  • Sine, \(\sin(x)\)
    • ctx.sin()
  • Cosine, \(\cos(x)\)
    • ctx.cos()
  • Tangent, \(\tan(x)\)
    • ctx.tan()
  • Cotangent, \(\cot(x)\)
    • ctx.cot()
  • Cosecant, \(\mathrm{csc}(x)\)
    • ctx.csc()
  • Secant, \(\mathrm{sec}(x)\)
    • ctx.sec()
  • Cardinal sine, \(\mathrm{sinc}(x)\)
    • ctx.sinc()
• Trigonometric functions, in multiples of \(\pi\)
  • Auxiliary function \(\mathrm{sinpi}(x) = \sin(\pi x)\)
    • ctx.sinpi()
  • Auxiliary function \(\mathrm{cospi}(x) = \cos(\pi x)\)
    • ctx.cospi()
  • Auxiliary function \(\mathrm{tanpi}(x) = \tan(\pi x)\)
    • ctx.tanpi()
  • Auxiliary function \(\mathrm{cotpi}(x) = \cot(\pi x)\)
    • ctx.cotpi()
  • Auxiliary function \(\mathrm{cscpi}(x) = \mathrm{csc}(\pi x)\)
    • ctx.cscpi()
  • Auxiliary function \(\mathrm{secpi}(x) = \mathrm{sec}(\pi x)\)
    • ctx.secpi()
  • Auxiliary function \(\mathrm{sincpi}(x) = \mathrm{sinc}(\pi x)\)
    • ctx.sincpi()
• Hyperbolic functions
  • Hyperbolic sine, \(\mathrm{sinh}(x)\)
    • ctx.sinh()
  • Hyperbolic cosine, \(\mathrm{cosh}(x)\)
    • ctx.cosh()
  • Hyperbolic tangent, \(\mathrm{tanh}(x)\)
    • ctx.tanh()
  • Hyperbolic cotangent, \(\mathrm{coth}(x)\)
    • ctx.coth()
  • Hyperbolic cosecant, \(\mathrm{csch}(x)\)
    • ctx.csch()
  • Hyperbolic secant, \(\mathrm{sech}(x)\)
    • ctx.sech()
• Inverse trigonometric functions, in radians
  • Inverse sine, \(\mathrm{asin}(x)\)
    • ctx.asin()
  • Inverse cosine, \(\mathrm{acos}(x)\)
    • ctx.acos()
  • Inverse tangent, \(\mathrm{atan}(x)\)
    • ctx.atan()
  • Inverse tangent, 2 arguments, \(\mathrm{atan2}(y, x)\)
    • ctx.atan2()
  • Inverse cotangent, \(\mathrm{acot}(x)\)
    • ctx.acot()
  • Inverse cosecant, \(\mathrm{acsc}(x)\)
    • ctx.acsc()
  • Inverse secant, \(\mathrm{asec}(x)\)
    • ctx.asec()
• Inverse hyperbolic functions
  • Inverse hyperbolic sine, \(\mathrm{asinh}(x)\)
    • ctx.asinh()
  • Inverse hyperbolic cosine, \(\mathrm{acosh}(x)\)
    • ctx.acosh()
  • Inverse hyperbolic tangent, \(\mathrm{atanh}(x)\)
    • ctx.atanh()
  • Inverse hyperbolic cotangent, \(\mathrm{acoth}(x)\)
    • ctx.acoth()
  • Inverse hyperbolic secant, \(\mathrm{asech}(x)\)
    • ctx.asech()
  • Inverse Hyperbolic Cosecant, \(\mathrm{acsch}(x)\)
    • ctx.acsch()
• Factorials, Gamma and related functions
  • Gamma function, \(\Gamma(x)\)
    • ctx.gamma()
  • Auxiliary function \(\Gamma(x+1)-1\)
    • ctx.real_gamma1pm1()
  • Log-gamma function, \(\log\Gamma(x)\)
    • ctx.lgamma()
  • Reciprocal Gamma function, \(1/\Gamma(x)\)
    • ctx.rgamma()
  • Factorial, \(x!\)
    • ctx.factorial()
  • Double factorial, \(x!!\)
    • ctx.double_factorial()
  • Rising factorial \(a^{\overline{n}} = (a)_n\)
    • ctx.rising_factorial()
  • Falling factorial, \((a)^{\underline{n}} = (a-n+1)^{\overline{n}}\)
    • ctx.falling_factorial()
  • Ratio of gamma functions, \(\Gamma(a)/\Gamma(b)\)
    • ctx.real_gamma_ratio()
  • Gamma-delta ratio, \(\Gamma(a)/\Gamma(a + \delta)\)
    • ctx.real_gamma_delta_ratio()
  • Beta function, \(B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a + b)\)
    • ctx.beta()
  • Binomial coefficient, \({}_nC_k = (k \cdot B(k, n-k+1))^{-1}\)
    • ctx.binomial()
• Miscellaneous functions
  • Lambert \(W\) function, \(W_0(x)\)
    • ctx.lambert_w0()
  • Lambert \(W\) function, \(W_{-1}(x)\)
    • ctx.lambert_wm1()
  • Lambert \(W\) (general case)
    • ctx.lambert_wk()
  • Arithmetic-geometric mean
    • ctx.agm()