## Fresnel integrals

FRESNEL(X) calculates the values of the Fresnel integrals for real values of vector X,

i.e.

C = \int_0^x cos(pi*t^2/2) dt, (0a)

S = \int_0^x sin(pi*t^2/2) dt (0b)

Also, it evaluates the following variations of the Fresnel integrals

C1 = \sqrt(2/pi) \int_0^x cos(t^2) dt, (1a)

S1 = \sqrt(2/pi) \int_0^x sin(t^2) dt (1b)

and

C2 = \sqrt(1/2/pi) \int_0^x cos(t) / \sqrt(t) dt, (2a)

S2 = \sqrt(1/2/pi) \int_0^x sin(t) / \sqrt(t) dt (2b)

The integrals are calculated as follows:

- Values of X in the interval [-5,5] are looked up in Table 7.7 in [1] and interpolated ('cubic'), if necessary. This table has values of the Fresnel integrals with 7 significant digits and even linear interpolation gives an error of no more than 3e-004.

- Values outside the interval [-5,5] are evaluated using the approximations under Table 7.7 in [1]. The error is less than

3e-007.

NOTE: The Tables were OCR'ed and although thoroughly checked, there might be some mistakes. Please let me know if you find any.

REFERENCE

[1] Abramowitz, M. and Stegun, I. A. (Eds.). "Error Function and Fresnel Integrals." Ch. 7 in "Handbook of mathematical functions with formulas, Graphs, and mathematical tables", 9th printing. New York: Dover, pp. 295-329, 1970.

EXAMPLES OF USAGE

>> Y = fresnel(X,'c');

returns the Fresnel C integral according to eq. (0a)

>> Y = fresnel(X,'s',2);

returns the Fresnel S integral values according to eq. (2b)

>> [C,S] = fresnel(X,[],1)

returns the both the Fresnel C and S integral values according to eqs. (1a), (1b)

>> [C,S] = fresnel(X,[])

returns the both the Fresnel C and S integral values according to eqs. (0a), (0b)

### Cite As

Christos Saragiotis (2021). Fresnel integrals (https://www.mathworks.com/matlabcentral/fileexchange/20052-fresnel-integrals), MATLAB Central File Exchange. Retrieved .

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**Inspired:**
FresnelS and FresnelC

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