Explicit derivatives for complex analytic functions

src/dual.jl

@@ -238,13 +238,30 @@ function unary_dual_definition(M, f)
         val = $Mf(x)
         deriv = $(DiffRules.diffrule(M, f, :x))
     end)
-    return quote
+    real_diff_exp = quote
         @inline function $M.$f(d::$FD.Dual{T}) where T
             x = $FD.value(d)
             $work
             return $FD.dual_definition_retval(Val{T}(), val, deriv, $FD.partials(d))
         end
     end
+    if (M, f) in ((:Base, :abs), (:Base, :abs2), (:Base, :inv))
+        return real_diff_exp
+    else
+        complex_diff_expr = quote
+            @inline function $M.$f(d::Complex{<:$FD.Dual{T}}) where{T}
+                x = complex($FD.value(real(d)), $FD.value(imag(d)))
+                $work
+                re_deriv, im_deriv = reim(deriv)
+                re_partials = $FD.partials(real(d))
+                im_partials = $FD.partials(imag(d))
+                re_retval = $FD.dual_definition_retval(Val{T}(), real(val), re_deriv, re_partials, -im_deriv, im_partials)
+                im_retval = $FD.dual_definition_retval(Val{T}(), imag(val), im_deriv, re_partials, re_deriv, im_partials)
+                return complex(re_retval, im_retval)
+            end
+        end
+        return Expr(:block, real_diff_exp, complex_diff_expr)
+    end
 end
 
 function binary_dual_definition(M, f)
@@ -709,6 +726,22 @@ end
     Dual{Tz}(muladd(x, y, value(z)), partials(z))      # z_body
 )
 
+# inv(Complex) #
+#--------#
+
+function Base.inv(d::Complex{<:Dual{T}}) where{T}
+    FD = ForwardDiff
+    x = complex(FD.value(real(d)), FD.value(imag(d)))
+    val = inv(x)
+    deriv = - val * val
+    re_deriv, im_deriv = reim(deriv)
+    re_partials = FD.partials(real(d))
+    im_partials = FD.partials(imag(d))
+    re_retval = FD.dual_definition_retval(Val{T}(), real(val), re_deriv, re_partials, -im_deriv, im_partials)
+    im_retval = FD.dual_definition_retval(Val{T}(), imag(val), im_deriv, re_partials, re_deriv, im_partials)
+    return complex(re_retval, im_retval)
+end
+
 # sin/cos #
 #--------#
 
@@ -727,6 +760,32 @@ end
     return (Dual{T}(sd, cd * partials(d)), Dual{T}(cd, -sd * partials(d)))
 end
 
+function Base.sin(d::Complex{<:Dual{T}}) where{T}
+    FD = ForwardDiff
+    x = complex(FD.value(real(d)), FD.value(imag(d)))
+    val = sin(x)
+    deriv = cos(x)
+    re_deriv, im_deriv = reim(deriv)
+    re_partials = FD.partials(real(d))
+    im_partials = FD.partials(imag(d))
+    re_retval = FD.dual_definition_retval(Val{T}(), real(val), re_deriv, re_partials, -im_deriv, im_partials)
+    im_retval = FD.dual_definition_retval(Val{T}(), imag(val), im_deriv, re_partials, re_deriv, im_partials)
+    return complex(re_retval, im_retval)
+end
+
+function Base.cos(d::Complex{<:Dual{T}}) where{T}
+    FD = ForwardDiff
+    x = complex(FD.value(real(d)), FD.value(imag(d)))
+    val = cos(x)
+    deriv = -sin(x)
+    re_deriv, im_deriv = reim(deriv)
+    re_partials = FD.partials(real(d))
+    im_partials = FD.partials(imag(d))
+    re_retval = FD.dual_definition_retval(Val{T}(), real(val), re_deriv, re_partials, -im_deriv, im_partials)
+    im_retval = FD.dual_definition_retval(Val{T}(), imag(val), im_deriv, re_partials, re_deriv, im_partials)
+    return complex(re_retval, im_retval)
+end
+
 # sincospi #
 #----------#
 

test/DerivativeTest.jl

@@ -113,4 +113,11 @@ end
     @test ForwardDiff.derivative(x -> (1+im)*x, 0) == (1+im)
 end
 
+@testset "analytic functions" begin
+    dexp(x) = ForwardDiff.derivative(y -> exp(complex(0, y)), x)
+    @test ForwardDiff.derivative(dexp, 0.0) ≈ -1
+    @test ForwardDiff.derivative(x -> exp(1im*x), 0.7) ≈ im * cis(0.7)
+    @test ForwardDiff.derivative(x -> sqrt(im + (1+im) * x), 1.23) ≈ (1+im) / (2 * sqrt(im + (1+im)*1.23))
+end
+
 end # module