Higher derivative gives NAN

[gkeee]

I want to compute the higher derivatives of f1(x) = exp(im * 2π * x) / (im * x). For example 8th derivative, ı used nth_derivative function with ForwardDiff, but 8th derivatives gives NAN. I changed the function as f2(x) = exp(im * 2π * x) * (im * x)^(-1). f2 function given NAN if I choose x=1/1000. I changed the function as f3(x) = -im * exp(im * 2π * x) * x^(-1), it is calculated. But the nth_derivative so slow.

I don’t understand the effect of the functions and how can I compute fast the nth derivative of exp(im * 2π * x) / (im * x) ?

using ForwardDiff

function nth_derivative(f, x, n)
    if n == 0
        return f(x)
    else
        return ForwardDiff.derivative(x -> nth_derivative(f, x, n - 1), x)
    end
end

f3(x) = -im * exp(im * 2π * x) * x^(-1)
nth_derivative(f3, 1/1000, 8)

[nsajko]

First things first, take a look at:

Please read: make it easier to help you

Welcome to the Julia Discourse! We are enthusiastic about helping Julia programmers, both beginner and experienced. This public service announcement (PSA) outlines best practices when asking for help. Following these points makes it easier for us to help you and more likely you’ll get a prompt, useful answer. Keywords are highlighted to make it easier to refer to specific points. Choose a descriptive title that captures the key part of your question, eg “plots with multiple axes” instead of …

In particular, please use code block Markdown to make your message readable.

[Maysam_Gholampour]

I think you can use TaylorDiff.jl. this is a MWE.

using TaylorDiff

x = 0.1
derivative(sin, x, 10) # scalar derivative
v, direction = [3.0, 4.0], [1.0, 0.0]
derivative(x -> sum(exp.(x)), v, direction, 2) # directional derivative

[gkeee]

Taylor series does not give a result for my function

function f(w)
    return -im * exp(im * 2π * w) / w^
end

ı try

derivative(f,1,3)

It give “false”.

[Maysam_Gholampour]

please first read :
Please read: make it easier to help you

If x is real, you can simplify f(x) to f(x) = h(x) + i * q(x), where h(x) and q(x) are real-valued functions, and i is the imaginary unit.

To derive f(x) into its real and imaginary parts separately:

1. Real Part:

   • Re(f(x)) = h(x)

2. Imaginary Part:

   • Im(f(x)) = q(x)

When differentiating f(x) with respect to x, and treating the real and imaginary parts separately:

• The derivative of f(x) is:

  • d/dx [f(x)] = d/dx [h(x) + i * q(x)] = dh(x)/dx + i * dq(x)/dx

• Separating the derivative into real and imaginary parts:

  • d/dx [f(x)] = dh(x)/dx + i * dq(x)/dx

Here, dh(x)/dx is the derivative of the real part h(x) with respect to x, and dq(x)/dx is the derivative of the imaginary part q(x) with respect to x.