The file FresnelIntegrals.mth defines functions for approximating Fresnel integral and related integrals.  These integrals are described in Chapter 7 of Abramowitz and Stegun [1965].  See the introduction to ExponentialIntegrals.mth7RN4D for a discussion of accuracy.  The function definitions in the file are automatically loaded when any of its functions are first used.


DAWSON(x) simplifies to Dawson’s integral F(x) defined in terms of the complex error function (see Error FunctionsError_Functions) as −√π/2·i·exp(−x²)·erf(i·x).


FRESNEL_SIN(z) approximates to the Fresnel sine integral S(z) defined as the integral of sin(π·t²/2) from t=0 to z.


FRESNEL_SIN_SERIES(z, m) approximates to m+1 terms of a series for the above Fresnel sine integral.


FRESNEL_SIN_J(z, m) approximates to the Fresnel sine integral S(z) based on the sum of m+1 spherical Bessel functions of the first kind using formula 7.3.16 of the Handbook of Mathematical Functions by Milton Abramowitz and Irene A. Stegun.  Larger values of m give more accurate results.


FRESNEL_SIN_ASYMP(z) simplifies to a five-term asymptotic approximation of the above Fresnel sine integral for large magnitude z.


FRESNEL_COS(z) approximates to the Fresnel cosine integral C(z) defined as the integral of cos(π·t²/2) from t=0 to z.


FRESNEL_COS_SERIES(z, m) approximates m+1 terms of a series for the above Fresnel cosine integral.


FRESNEL_COS_J(z, m) approximates to the Fresnel cosine integral C(z) based on the sum of m+1 spherical Bessel functions of the first kind using formula 7.3.15 of the Handbook of Mathematical Functions by Milton Abramowitz and Irene A. Stegun.  Larger values of m give more accurate results.


FRESNEL_COS_ASYMP(z) simplifies to a five-term asymptotic approximation of the above Fresnel cosine integral for large magnitude z.


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