Fixed factor in derivative of Kaiser windows; test using AD

src/arithmetic.jl

@@ -685,7 +685,7 @@ function window_derivative(w::Kaiser, x)
     abs(2x) > 1 && return zero(x)
     a = π*w.α
     f = √(1 - (2x)^2)
-    pf = 2a*x/(besseli(0,a)*f)
+    pf = 4a*x/(besseli(0,a)*f)
     -pf*besseli(1, a*f)
 end
 

test/apodized_fields.jl

@@ -85,7 +85,8 @@
                                                 ElectricFields.BlackmanHarris(),
                                                 ElectricFields.Kaiser(3), ElectricFields.Kaiser(2),
                                                 ElectricFields.Rect(), Zero(), One())
-        w = ElectricFields.window_value.(window, x)
+        wx = Base.Fix1(ElectricFields.window_value, window)
+        w = wx.(x)
         ∂w = ElectricFields.window_derivative.(window, x)
 
         @test all(iszero, w[pre_sel])
@@ -103,6 +104,9 @@
             # the derivative, just that the behaviour is reasonable.
             @test all(≥(0), ∂w[mid_sel1])
             @test all(≤(0), ∂w[mid_sel2])
+            # Here we however compare the analytic derivatives with
+            # AD.
+            test_approx_eq(ForwardDiff.derivative.(wx, x[2:end-1]), ∂w[2:end-1])
         end
 
         @test all(iszero, w[post_sel])
@@ -135,7 +139,7 @@
     # For the same reason, the intensity is not exactly the original
     # intensity (flat for this particular field) times the window
     # value squared, hence the high tolerance.
-    @test abs2.(w[mid_sel]) ≈ Iwv[mid_sel] rtol=5e-2
+    test_approx_eq(abs2.(w[mid_sel]), Iwv[mid_sel], rtol=0.2)
     @test all(iszero, Fwv[post_sel])
 
     @test all(iszero, Awv[pre_sel])

test/runtests.jl

@@ -5,6 +5,7 @@ using UnitfulAtomic
 using LinearAlgebra
 using IntervalSets
 using FFTW
+using ForwardDiff
 using PrettyTables
 
 function test_approx_eq(a, b; on_fail::Union{Nothing,Function}=nothing, isbroken=false, kwargs...)