fresnel


fresnel, an Octave code which evaluates the Fresnel cosine and sine integrals.

The Fresnel integrals are defined as:

      c(x) = Integral ( 0 <= t <= x ) cos(pi/2 t^2) dt
      s(x) = Integral ( 0 <= t <= x ) sin(pi/2 t^2) dt
(Some definitions omit the factor of pi/2.)

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

fresnel is available in a MATLAB version and an Octave version.

Related Data and Programs:

fresnel_test

gaussian, an Octave code which evaluates the Gaussian function for arbitrary mu and sigma, its antiderivative, and derivatives of arbitrary order.

humps, an Octave code which evaluates the humps() function, its first and second derivatives, and its antiderivative. The functions are plotted.

runge, an Octave code which evaluates the Runge() function, its first and second derivatives, and its antiderivative. This function causes a breakdown for polynomial interpolation over equally spaced nodes in [-1,+1]. The functions are plotted.

sigmoid, an Octave code which evaluates the sigmoid function s(x)=1/(1+exp(-x)) or its derivatives of any order. The test code creates graphic images.

sinc, an Octave code which evaluates the sinc() function, its first and second derivative and its antiderivative. The normalized function is defined by sincn=sin(pi*x)/(pi*x), the unnormalized function is sincu=sin(x)/x. Plots of these functions are made.

steinerberger, an Octave code which evaluates the Steinerberger function, a continuous function with discontinuous derivative, which is very hard to accurately plot, integrate, minimize, or interpolate.

Reference:

  1. John D Cook,
    Cornu's spiral,
    Posted 23 March 2026.
    https://www.johndcook.com/blog/2016/03/23/cornus-spiral/
  2. Shanjie Zhang, Jianming Jin,
    Computation of Special Functions,
    Wiley, 1996,
    ISBN: 0-471-11963-6,
    LC: QA351.C45.

Source Code:


Last revised on 08 July 2025.