riemann_zeta.md

riemann_zeta

• cmath[meta header]
• function[meta id-type]
• std[meta namespace]
• [mathjax enable]
• cpp17[meta cpp]

namespace std {
float riemann_zetaf(float x);
double riemann_zeta(double x);
long double riemann_zetal(long double x);
}

概要

リーマンのゼータ関数 (Riemann zeta function) を求める。

戻り値

引数 x のリーマンゼータ関数 $$ \zeta (x) = \begin{cases} \displaystyle \sum_{k=1}^\infty k^{-x} & \text{for } x > 1 \ \displaystyle \frac{1}{1 - 2^{1-x}} \sum_{k=1}^\infty (-1)^{k-1} k^{-x} & \text{for } 0 \le x \le 1 \ \displaystyle 2^x \pi^{x-1} \sin \frac{\pi x}{2} \Gamma (1 - x) \zeta(1 - x) & \text{for } x < 0 \end{cases} $$ を返す。

例

#include <cmath>
#include <iostream>

int main() {
  std::cout << "riemann_zeta(-2) = " << std::riemann_zeta(-2) << "\n";  // 0
  std::cout << "riemann_zeta(-1) = " << std::riemann_zeta(-1) << "\n";  // -1 / 12
  std::cout << "riemann_zeta(0)  = " << std::riemann_zeta(0) << "\n";   // -1 / 2
  std::cout << "riemann_zeta(1)  = " << std::riemann_zeta(1) << "\n";   // ∞
  std::cout << "riemann_zeta(2)  = " << std::riemann_zeta(2) << "\n";   // π^2 / 6
}

• std::riemann_zeta[color ff0000]

出力例

riemann_zeta(-2) = 0
riemann_zeta(-1) = -0.0833333
riemann_zeta(0)  = -0.5
riemann_zeta(1)  = inf
riemann_zeta(2)  = 1.64493

バージョン

言語

• C++17

処理系

• Clang: ??
• GCC: 7.1.0
• ICC: ??
• Visual C++: ??

参照

• N3060 JTC1.22.29124 Programming Language C++ — Special Math Functions
• WG21 P0226R1 Mathematical Special Functions for C++17, v5
• ISO/IEC 29124:2010 Information technology -- Programming languages, their environments and system software interfaces -- Extensions to the C++ Library to support mathematical special functions