jn

constexpr Complex jn(const int n, const Complex &z) noexcept: Evaluates the Bessel function of the first kind [1] for a complex input and integer order.

Parameters

const int n

An integer.

const Complex &z

A complex number.

Returns

type Complex

A complex number.

The Bessel functions of the first kind are the solutions to the following differential equation:

\[x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2) = 0\]

For an integer order \(n\), the following integral representation [2] can be used:

\[J_n(z) = \frac{1}{\pi}\int_{0}{\pi}\cos(z\sin(\theta) - n\theta)d\theta\]

Example

int n = 1;
Complex z = 1_j;
std::cout << jn(n, z) << "\n";

Output:

1.87315e-17 + 0.565159j

References