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SandMath_44 Manual Fresnel Integrals.
Fresnel integrals, S(x) and C(x), are two transcendental functions named after Augustin-Jean Fresnel
that are used in optics. They arise in the description of near field Fresnel diffraction phenomena, and
are defined through the following integral representations:
The function CSX will calculate both S(x) and C(x) for the argument in X, returning the results in Y and
X respectively. It is a short FOCAL program that uses (yes you guessed it) the Generalized Hypergeometric function, according to the expressions:
S(x) = ( π x3/6 ) 1F2( 3/4 ; 3/2 , 7/4 ; -π2 x4/16 ),
and
C(x) = x 1F2( 1/4 ; 1/2 , 5/4 ; -π2 x4/16 )
The figure below shows both functions plotted for 0<x<5
REGISTERS:
FLAGS:
R00 thru R04
none
Examples:
1.5 ΣFL$ "CSX" ->
4 ΣFL$ "CSX" ->
Stack
Y
X
C(1.5) = 0.445261176
C(4) = 0.498426033
Input
n/a
x
X<>Y,
X<>Y,
Output
S(x)
C(x)
S(1.5) = 0.697504960
S(4) = 0.420515754
Or: [ΣFL], [H], [C]
(c) Ángel M. Martin Revision 44_E Page 80