integral

constexpr Complex integral(Complex (*f)(Complex, const std::vector<Complex (*)(Complex)> &fargs, std::vector<Complex>), const Complex &a, const Complex &b) noexcept

constexpr Complex integral(Complex (*f)(Complex, const std::vector<Complex (*)(Complex)> &fargs, std::vector<Complex>), const Complex &a, const Complex &b, const std::vector<Complex (*)(Complex)> &fargs) noexcept

constexpr Complex integral(Complex (*f)(Complex, const std::vector<Complex (*)(Complex)> &fargs, std::vector<Complex>), const Complex &a, const Complex &b, const std::vector<Complex> &args) noexcept

constexpr Complex integral(Complex (*f)(Complex, const std::vector<Complex (*)(Complex)> &fargs, std::vector<Complex>), const Complex &a, const Complex &b, const std::vector<Complex (*)(Complex)> &fargs, const std::vector<Complex> &args) noexcept: Evaluates the integral of a complex function at a point \(z\) from integral bounds \(a\) to \(b\). The integral is implemented using 15-point Gauss-Kronrod quadrature [1].

Parameters

Complex (*f)(Complex)

The function to differentiate.

const Complex &z

A complex number.

const Complex &a

Lower integration bound. Use NINF from <Constants.hpp> for \(-\infty\).

const Complex &b

Upper integration bound. Use INF from <Constants.hpp> for \(\infty\).

const std::vector<Complex (*)(Complex)> &fargs

Optional list of function arguments if needed for use by the integrand.

const std::vector<Complex> &args

Optional list of variable arguments if needed for use by the integrand.

Returns

type Complex

A complex number.

This function evaluates the following definite integral of a function \(f\):

\[\int_{a}^{b} f(z) dz\]

To approxiate the integral, the following numerical quadrature method is used:

\[\int_{a}^{b} f(z) dz \approx \sum_{i = 1}^{n}w_if(z_i)\]

for some pre-tabulated Kronrod weights \(w\) and evaluation points \(z\).

Example

auto fn = [](Complex z) { return z*z }; // z^2
std::cout << integral(fn, 0, 1) << "\n"; // z^3/3 |0 -> 1 = 1/3

Output:

0.333333 + 0j

References