yv
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constexpr Complex yv(const double v, const Complex &z) noexcept
Evaluates the Bessel function of the second kind [1] for a complex input and real order.
Parameters
- const double v
A real number.
- const Complex &z
A complex number.
Returns
- type Complex
A complex number.
The Bessel functions of the second kind are the solutions to the following differential equation:
\[x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + (x^2 - v^2)y = 0\]
For a real order \(v\), the following integral representation [2] can be used:
\[Y_v(z) = \frac{2(\frac{1}{2}z)^v}{\pi^\frac{1}{2}\Gamma(v + \frac{1}{2})}(\int_{0}^{1}(1 - t^2)^{v - \frac{1}{2}}\sin(zt)dt - \int_{0}^{\infty}e^{-zt}(1 + t^2)^{v - \frac{1}{2}}dt)\]
for \(\Re(v) > -\frac{1}{2}\) and \(\DeclareMathOperator\arg{arg} |\arg(z)| < \frac{1}{2}\pi\).
Example
int n = 0.5;
Complex z = 1 + 1_j;
std::cout << yv(n, z) << "\n";
Output:
-0.261856 + 0.828685j
References