Spiced up readme

Author: kiranshila

README.md

@@ -4,3 +4,47 @@
 [![Dev](https://img.shields.io/badge/docs-dev-blue.svg)](https://kiranshila.github.io/FresnelIntegrals.jl/dev)
 [![Build Status](https://travis-ci.com/kiranshila/FresnelIntegrals.jl.svg?branch=master)](https://travis-ci.com/kiranshila/FresnelIntegrals.jl)
 [![Codecov](https://codecov.io/gh/kiranshila/FresnelIntegrals.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/kiranshila/FresnelIntegrals.jl)
+
+A quick Julia library for calculating Fresnel Integrals using the error function from [SpecialFunctions.jl](https://github.com/JuliaMath/SpecialFunctions.jl).
+
+```julia
+using FresnelIntegrals
+using Plots
+
+z = -30:0.001:30
+plot(x->real(fresnelc(x)), x->real(fresnels(x)),z,legend=false,xlabel="C(x)",ylabel="S(x)",title="Euler Spiral")
+```
+![Spiral](Spiral.png)
+
+As expected, this is much faster than the numeric integral solution
+
+```julia
+using QuadGK
+using FresnelIntegrals
+
+julia> @benchmark fresnelc(1.8)
+BenchmarkTools.Trial:
+  memory estimate:  0 bytes
+  allocs estimate:  0
+  --------------
+  minimum time:     410.815 ns (0.00% GC)
+  median time:      412.035 ns (0.00% GC)
+  mean time:        426.088 ns (0.00% GC)
+  maximum time:     1.017 μs (0.00% GC)
+  --------------
+  samples:          10000
+  evals/sample:     200
+
+julia> @benchmark quadgk(z->cos(π*z^2/2),0,1.8)
+BenchmarkTools.Trial:
+  memory estimate:  1.00 KiB
+  allocs estimate:  42
+  --------------
+  minimum time:     2.181 μs (0.00% GC)
+  median time:      2.250 μs (0.00% GC)
+  mean time:        3.357 μs (31.54% GC)
+  maximum time:     7.919 ms (99.89% GC)
+  --------------
+  samples:          10000
+
+```

Spiral.png

@@ -2,6 +2,7 @@ module FresnelIntegrals
 using SpecialFunctions
 export fresnelc
 export fresnels
+export fresnel
 
 """
         fresnelc(z)
@@ -15,4 +16,10 @@ Calculates the Fresnel sine integral for the number z
 """
 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))
+
 end # module

src/FresnelIntegrals.jl

@@ -1,6 +1,13 @@
 using FresnelIntegrals
 using Test
+using QuadGK
 
 @testset "FresnelIntegrals.jl" begin
-    # Write your own tests here.
+    # Generate random complex number
+    z = randn(ComplexF64)
+    # Test by comparing to numeric solution
+    @test fresnelc(z) ≈ quadgk(t->cos(π*t^2/2),0,z)[1]
+    @test fresnels(z) ≈ quadgk(t->sin(π*t^2/2),0,z)[1]
+    # Test just for code coverage 😄
+    @test (fresnelc(z),fresnels(z)) == fresnel(z)
 end