Exponentially Scaled Complementary Error Function
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of the Complementary Error Function Keywords: error functions, repeated integrals of the complementary error function Referenced by: §12.7(ii) Permalink: http://dlmf.nist.gov/7.18 See also: info for 7 Contents §7.18(i) error function integral Definition §7.18(ii) Graphics §7.18(iii) Properties §7.18(iv) Relations to Other Functions §7.18(v)
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Continued Fraction §7.18(vi) Asymptotic Expansion §7.18(i) Definition Keywords: repeated integrals of the complementary error function Permalink: error function table http://dlmf.nist.gov/7.18.i See also: info for 7.18 7.18.1 i-1erfc(z) =2πe-z2, i0erfc(z) =erfcz, Symbols: erfcz: complementary error function, e: base of exponential function, inerfc(z): repeated integrals of the complementary erf(inf) error function and z: complex variable A&S Ref: 7.2.1 (in modified form) Referenced by: §7.22(iii) Permalink: http://dlmf.nist.gov/7.18.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: info for 7.18(i) and for n=0,1,2,…, 7.18.2 inerfc(z)=∫z∞in-1erfc(t)dt=2π∫z∞(t-z)nn!e-t2dt. Defines: inerfc(z): repeated integrals of the complementary error function Symbols: dx: differential of x, e: base of exponential function,
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!: factorial (as in n!), ∫: integral, z: complex variable and n: nonnegative integer A&S Ref: 7.2.1 (in modified form) 7.2.3 Permalink: http://dlmf.nist.gov/7.18.E2 Encodings: TeX, pMML, png See also: info for 7.18(i) §7.18(ii) Graphics Notes: These graphs were produced at NIST. Keywords: repeated integrals of the complementary error function Permalink: http://dlmf.nist.gov/7.18.ii See also: info for 7.18 Figure 7.18.1: Repeated integrals of the scaled complementary error function 2nΓ(12n+1)inerfc(x), n=0,1,2,4,8,16. Symbols: Γ(z): gamma function, inerfc(z): repeated integrals of the complementary error function, x: real variable and n: nonnegative integer Keywords: repeated integrals of the complementary error function Permalink: http://dlmf.nist.gov/7.18.F1 Encodings: pdf, png See also: info for 7.18(ii) §7.18(iii) Properties Notes: See Hartree (1936). Keywords: derivatives, repeated integrals of the complementary error function Permalink: http://dlmf.nist.gov/7.18.iii See also: info for 7.18 7.18.3 ddzinerfc(z)=-in-1erfc(z), n=0,1,2,…, Symbols: dfdx: derivative of f with respect to x, inerfc(z): repeated integrals of the complementary error function, z: complex variable and n: nonnegative integer A&S Ref: 7.2.8 Permalink: http://dlmf.nist.gov/7.18.E3 E
Syntax:RESULT = ERFC_SCALED(X) Arguments: X The type shall be https://gcc.gnu.org/onlinedocs/gfortran/ERFC_005fSCALED.html REAL. Return value:The return value is of type REAL and of the same kind as X. Example: program test_erfc_scaled real(8) :: x = 0.17_8 x = erfc_scaled(x) end program test_erfc_scaled
control functions Polynomial, Vector, Statistic Operations Argument reduction for trigonometric functions Basic double-extended (double-double) functions Other functions Special functions http://www.wolfgang-ehrhardt.de/amath_functions.html Bessel functions and related Elliptic integrals, elliptic / theta http://www.roguewave.com/products-services/imsl-numerical-libraries/functional-catalog/mathematical-special-functions/error-functions-and-related-functions functions Error function and related Exponential integrals and related Gamma function and related Orthogonal polynomials, Legendre functions, and related Hypergeometric functions and related Statistical distributions Zeta functions, polylogarithms, and related Other special functions AMTools and DAMTools functions Function error function minimization Zero finding Numerical integration Convergence acceleration Solving quadratic and cubic equations AMath and DAMath complex functions Complex arithmetic and basic functions Complex transcendental functions References Introduction The AMath package contains Pascal/Delphi source for accurate mathematical methods without using multi precision arithmetic; it is designed for the 80-bit exponentially scaled complementary extended data type. DAMath makes most of the AMath functions available for 64-bit systems without extended precision or 387-FPU. Please note that the high accuracy can only be achieved with the rmNearest rounding mode; it decreases if other modes are used. The test suites run without 'failure warnings about relative errors' on Intel CPUs on Win98, Win2000, WinXP, and Win7. There may be some sporadic warnings with other processors or operating systems; normally, these are not AMath bugs but features of the CPU (and can be avoided by using slightly increased error levels). Note that this does not mean, that the Intel CPUs are 'better', it only reflects the development environment where the error bounds are defined; if developed on other systems, some warnings could occur for Intel CPUs. Note for FreePascal users: The development versions 3.0.1, 3.1.1 but also the
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