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## Special and complex-valued functions

Gnuplot 6 provides an expanded set of complex-valued functions and updated versions of some functions that were present in earlier versions.

- New: Riemann zeta function with complex domain and range. See zeta.
- Updated lower incomplete gamma function with improved domain and precision. Complex arguments accepted. See igamma.
- New upper incomplete gamma function (real arguments only). See uigamma.
- Updated incomplete beta function with improved domain and precision. See ibeta.
- New function for the inverse incomplete gamma function. See invigamma.
- New function for the inverse incomplete beta function. See invibeta.
- New complex function LambertW(z,k) returns the kth branch of multivalued function W_k(z).
  Note that the older function lambertw(x) = real(LambertW( real(z), 0 )). See LambertW.
- New complex function lnGamma(z). Note that existing function lgamma(x) = real(lnGamma(real(z)). See lnGamma.
- Complex function conj(z) returns the complex conjugate of z.
- Synchrotron function F(x), see SynchrotronF.
- acosh(z) domain extended to cover negative real axis.
- asin(z) asinh(z) improved precision for complex arguments.
- Predefined variable I = sqrt(-1) = {0,1} for convenience.
  This is useful because gnuplot does not accept {a,b} as a valid complex constant but does accept (a + b*I) as a valid complex expression.

Additional special functions are supported if a suitable external library is found at build time. See special_functions.

- Complex Bessel functions Iν(z), Jν(z), Kν(z), Yν(z) of order ν (real) with complex argument z. See BesselK.
- Complex Hankel functions H1ν(z), H2ν(z) of order ν with complex z. See BesselH1.
- Complex Airy functions Ai(z), Bi(z).
- Complex exponential integral of order n. See expint.
- Fresnel integrals C(x) and S(x). See FresnelC.
- Function VP_fwhm(sigma,gamma) returns the full width at half maximum of the Voigt profile. See VP, VP_fwhm.