Added Fresnel integrals #398

src/SpecialFunctions.jl

@@ -78,10 +78,14 @@ export
     expintx,
     sinint,
     cosint,
-    lbinomial
+    lbinomial,
+    fresnelc,
+    fresnels,
+    fresnel 
 
 include("bessel.jl")
 include("erf.jl")
+include("fresnel.jl")
 include("ellip.jl")
 include("expint.jl")
 include("sincosint.jl")

src/fresnel.jl

@@ -0,0 +1,25 @@
+
+
+# Code ported from https://github.com/kiranshila/FresnelIntegrals.jl
+
+
+"""
+        fresnelc(z)
+Calculates the Fresnel cosine integral for the number z for
+    ``C(z) = \\int_{0}^{z} \\cos{\\left(\\frac{\\pi t^2}{2}\\right)}dt``
+"""
+fresnelc(z::Number) = 0.25*(1-1im)*(1im*erf(0.5*(1-1im)*z*√(π)) + erf(0.5*(1+1im)*z*√(π)))
+
+"""
+        fresnels(z)
+Calculates the Fresnel sine integral for the number z for
+    ``S(z) = \\int_{0}^{z} \\sin{\\left(\\frac{\\pi t^2}{2}\\right)}dt``
+"""
+fresnels(z::Number) = 0.25*(1+1im)*(-1im*erf(0.5*(1-1im)*z*√(π)) + erf(0.5*(1+1im)*z*√(π)))
+
+"""
+        fresnel(z)
+Calculates the cosine and sine fresnel integrals
+"""
+fresnel(z::Number) = (fresnelc(z),fresnels(z))
+

test/fresnel.jl

@@ -0,0 +1,12 @@
+@testset "fresnel" begin
+    # Precise values come from WolframAlpha calculator
+    # One could add more decimals and more tests if needed
+
+    @test fresnels(1.)      ≈ 0.4382591473903
+    @test fresnelc(1.)      ≈ 0.7798934003768
+    @test fresnels(sqrt(2)) ≈ 0.7139722140219
+    @test fresnelc(sqrt(2)) ≈ 0.5288915951112
+
+    z = rand(ComplexF64)
+    @test fresnel(z) == (fresnelc(z),fresnels(z))
+end

test/runtests.jl

@@ -32,7 +32,8 @@ tests = [
     "gamma",
     "sincosint",
     "other_tests",
-    "chainrules"
+    "chainrules",
+    "fresnel"
 ]
 
 const testdir = dirname(@__FILE__)