# Fresnel Sine Integrals and Fresnel Cosine Integrals

 Function Library Home Fresnel Sin and Cos Integrals

## Table of the Fresnel Sine and Cosine Integrals

### The Fresnel Sine Integral

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

#### Function List

• double Fresnel_Sine_Integral( double x )

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

• long double xFresnel_Sine_Integral( long double x )

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

#### Source Code

C source code is available for these routines:
• The file, fresnel_sine_integral.c, contains the functions Fresnel_Sine_Integral( ) and xFresnel_Sine_Integral( ).

Dependencies: In addition to the file fresnel_sine_integral.c, the functions Fresnel_Sine_Integral( ) and xFresnel_Sine_Integral( ) require the following file:

### The Fresnel Cosine Integral

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

#### Function List

• double Fresnel_Cosine_Integral( double x )

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

• long double xFresnel_Cosine_Integral( long double x )

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

#### Source Code

C source code is available for these routines:

### Fresnel Auxiliary Sine Integral

There are several common, but non-equivalent, 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

• 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.

#### Source Code

C source code is available for these routines:
• The file, fresnel_auxiliary_sine_integral.c, contains the functions Auxiliary_Fresnel_Sine_Integral( ) and xAuxiliary_Fresnel_Sine_Integral( ).

Dependencies: In addition to the file fresnel_auxiliary_sine_integrals.c, the functions Auxiliary_Fresnel_Sine_Integral( ) and xAuxiliary_Fresnel_Sine_Integral( ) require the following file:

### Fresnel Auxiliary Cosine Integral

There are several common, but non-equivalent, 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
f(√(π / 2) x) = ∫0 exp(-2xt) cos( (π / 2) t 2 ) dt, x ≥ 0
and
f(x2) = √(2 / π) ∫0 exp(-2tx²) cos( t 2 ) dt, x ≥ 0.

#### Function List

• double Fresnel_Auxiliary_Cosine_Integral ( double x )

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

#### Source Code

C source code is available for these routines:
• The file, fresnel_auxiliary_cosine_integrals.c, contains the functions Auxiliary_Fresnel_Cosine_Integral( ) and xAuxiliary_Fresnel_Cosine_Integral( ).

Dependencies: In addition to the file fresnel_auxiliary_cosine_integrals.c, the functions Auxiliary_Fresnel_Cosine_Integral( ) and xAuxiliary_Fresnel_Cosine_Integral( ) require the following file: