Fresnel Sine Integrals and Fresnel Cosine Integrals

Table of the Fresnel Sine and Cosine Integrals

Fresnel Sine Integral - S(x) = √(2 / π) ∫₀^x sin( t ² ) dt.
Fresnel Cosine Integral - C(x) = √(2 / π) ∫₀^x cos( t ² ) dt.
Fresnel Auxiliary Sine Integral - g(x) = √(2 / π) ∫₀^∞ exp(-2xt) sin( t ² ) dt, x ≥ 0.
Fresnel Auxiliary Cosine Integral - f(x) = √(2 / π) ∫₀^∞ exp(-2xt) cos( t ² ) dt, x ≥ 0.

The Fresnel Sine Integral

There are several common definitions of the Fresnel sin integral S(x). The definition of the Fresnel sin integral programmed here is
S(x) = √(2 / π) ∫₀^x sin( t ² ) dt. Two other common definitions are
S(√(π / 2) x) = ∫₀^x sin( (π / 2) t ² ) dt and
S(x²) = √(1 / 2π) ∫₀^x sin( t ) / √t dt.

Function List - S

Functions:

double Fresnel_Sine_Integral( double x )

This function returns S(x) = √(2 / π) ∫₀^x sin( t ² ) dt.

long double xFresnel_Sine_Integral( long double x )

This function returns S(x) = √(2 / π) ∫₀^x sin( t ² ) dt.

fresnel_sine_integral.c

The Fresnel Cosine Integral

There are several common definitions of the Fresnel cos integral C(x). The definition of the Fresnel cos integral programmed here is
C(x) = √(2 / π) ∫₀^x cos( t ² ) dt. Two other common definitions are
C(√(π / 2) x) = ∫₀^x cos( (π / 2) t ² ) dt and
C(x²) = √(1 / 2π) ∫₀^x cos( t ) / √t dt.

Function List - C

Functions:

double Fresnel_Cosine_Integral( double x )

This function returns C(x) = √(2 / π) ∫₀^x cos( t ² ) dt.

long double xFresnel_Cosine_Integral( long double x )

This function returns C(x) = √(2 / π) ∫₀^x cos( t ² ) dt.

fresnel_cosine_integral.c

Fresnel Auxiliary Sine Integral

There are several common definitions of the Fresnel auxiliary sin integral g(x). The definition of the Fresnel auxiliary sin integral programmed here is
g(x) = √(2 / π) ∫₀^∞ exp(-2xt) sin( t ² ) dt, x ≥ 0. Two other common definitions are
g(√(π / 2) x) = ∫₀^∞ exp(-2xt) sin( (π / 2) t ² ) dt, x ≥ 0 and
g(x²) = √(2 / π) ∫₀^∞ exp(-2tx²) sin( t ² ) dt, x ≥ 0.

Function List - g

Functions:

double Auxiliary_Fresnel_Sine_Integral ( double x )

This function returns g(x) = √(2 / π) ∫₀^∞ exp(-2xt) sin( t ² ) dt. The argument x must be nonnegative.

long double xAuxiliary_Fresnel_Sine_Integral ( long double x )

This function returns g(x) = √(2 / π) ∫₀^∞ exp(-2xt) sin( t ² ) dt. The argument x must be nonnegative.

fresnel_auxiliary_sine_integral.c

Fresnel Auxiliary Cosine Integral

There are several common definitions of the Fresnel auxiliary cos integral f(x). The definition of the Fresnel auxiliary cos integral programmed here is
f(x) = √(2 / π) ∫₀^∞ exp(-2xt) cos( t ² ) dt, x ≥ 0. Two other common definitions are
g(√(π / 2) x) = ∫₀^∞ exp(-2xt) cos( (π / 2) t ² ) dt, x ≥ 0 and
g(x²) = √(2 / π) ∫₀^∞ exp(-2tx²) cos( t ² ) dt, x ≥ 0.

Function List - f

Functions:

double Fresnel_Auxiliary_Cosine_Integral ( double x )

This function returns g(x) = √(2 / π) ∫₀^∞ exp(-2xt) cos( t ² ) dt. The argument x must be nonnegative.

long double xFresnel_Auxiliary_Cosine_Integral ( long double x )

This function returns g(x) = √(2 / π) ∫₀^∞ exp(-2xt) cos( t ² ) dt. The argument x must be nonnegative.

fresnel_auxiliary_cosine_integrals.c