FresnelS¶

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description¶

FresnelS[z]

gives the Fresnel integral S(z) = Integral_0^z Sin[Pi t^2/2] dt.

FresnelS[+-Infinity] = +-1/2, FresnelS[+-I Infinity] = -+I/2.

Notes

An entire, odd function with no branch cuts. FresnelS\[0\] = 0, Real and complex inputs evaluate numerically at machine or arbitrary (MPFR) precision; D\[FresnelS\[z\], z\] = Sin\[Pi z^2/2\]. Listable.

Examples¶

No verified examples yet for this function.

Algorithm¶

Mathilda -- the Fresnel integrals (Pi/2-normalized).

  FresnelC[z] = Int_0^z Cos[Pi t^2 / 2] dt
  FresnelS[z] = Int_0^z Sin[Pi t^2 / 2] dt

Both are entire and odd, with no branch cuts. FresnelC and FresnelS share one numeric kernel: the pair (C, S) is computed together and each builtin returns its component. Evaluation is layered so each kind of argument takes the cheapest route:

  exact special values     ->  0, +-1/2, +-I/2, Indeterminate
  machine / arbitrary real ->  MPFR series (small |x|) or asymptotic (large)
  complex (any precision)  ->  the paired A/B ncpx series with guard bits
  everything else          ->  stays symbolic (return NULL)

Convergent Maclaurin series (valid for all z):

  C(z) = Sum_{m>=0} (-1)^m (Pi/2)^(2m)   z^(4m+1) / ((2m)!   (4m+1)),
  S(z) = Sum_{m>=0} (-1)^m (Pi/2)^(2m+1) z^(4m+3) / ((2m+1)! (4m+3)).

Equivalently A(z) = C(z) + i S(z) = Sum_{k>=0} (i Pi/2)^k z^(2k+1)/(k!(2k+1)) and B(z) = C(z) - i S(z) is the same with i -> -i; then C = (A+B)/2 and S = (A-B)/(2i). The real path sums C and S as two real series directly; the complex path sums A and B together in one ncpx loop.

The partial sums reach magnitude ~e^((Pi/2)|z|^2) before the O(1)-sized answer emerges, so the MPFR paths add ~(Pi/2)|z|^2/ln2 guard bits to absorb that cancellation exactly. For large real |x| the convergent series is infeasible; there we use the asymptotic expansion (DLMF 7.12)

  C(x) = 1/2 + f(x) sin(Pi x^2/2) - g(x) cos(Pi x^2/2),
  S(x) = 1/2 - f(x) cos(Pi x^2/2) - g(x) sin(Pi x^2/2),
    f(x) ~ (1/(Pi x))   Sum_j (-1)^j (4j-1)!! / (Pi x^2)^(2j),
    g(x) ~ (1/(Pi^2 x^3)) Sum_j (-1)^j (4j+1)!! / (Pi x^2)^(2j),

summed to the smallest term (optimal truncation). The asymptotic constant 1/2 is the value only in a sector around the real axis (Stokes phenomenon), so it is used for real inputs only; complex inputs always use the convergent series (correct everywhere, merely costlier for large |z|).

Attributes: Listable, NumericFunction, Protected.

Implementation notes¶

Attributes: Listable, NumericFunction, Protected.