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 3e004.
 Values outside the interval [5,5] are evaluated using the approximations under Table 7.7 in [1]. The error is less than
3e007.
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. 295329, 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 (2023). Fresnel integrals (https://www.mathworks.com/matlabcentral/fileexchange/20052fresnelintegrals), MATLAB Central File Exchange. Retrieved .
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Acknowledgements
Inspired: FresnelS and FresnelC
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Version  Published  Release Notes  

1.1.0.0  Corrected the OCR typos mentioned by Peter Volegov (thanks, Peter, for taking the time) and three or four others as well. I checked it with some other relevant functions and it seems to be working OK now (differences are of the order of 1e8).


1.0.0.0 