| from mpmath import mp | |
| mp.dps = 110 | |
| def _conv_trunc(a, b, n): | |
| res = [mp.mpf("0")] * n | |
| na = min(len(a), n) | |
| nb = min(len(b), n) | |
| for i in range(na): | |
| ai = a[i] | |
| if not ai: | |
| continue | |
| m = min(nb, n - i) | |
| for j in range(m): | |
| res[i + j] += ai * b[j] | |
| return res | |
| def _tail_asymptotic(X0, N=300): | |
| # Asymptotic series coefficients for K0(x): | |
| # K0(x) ~ sqrt(pi/(2x)) * exp(-x) * sum_{k>=0} c_k / x^k, x -> +inf | |
| # with recurrence (nu=0, mu=0): c_0=1, | |
| # c_k = c_{k-1} * (-(2k-1)^2) / (8k) | |
| c = [mp.mpf("0")] * N | |
| c[0] = mp.mpf("1") | |
| for k in range(1, N): | |
| c[k] = c[k - 1] * (-(2 * k - 1) ** 2) / (mp.mpf(8) * k) | |
| # I0(5x) asymptotic has series sum_{k>=0} (-1)^k c_k / (5x)^k | |
| p = [mp.mpf("0")] * N | |
| inv5 = mp.mpf(1) / 5 | |
| inv5pow = mp.mpf(1) | |
| for k in range(N): | |
| pk = c[k] * inv5pow | |
| if k & 1: | |
| pk = -pk | |
| p[k] = pk | |
| inv5pow *= inv5 | |
| # q = (sum c_k/x^k)^5 truncated | |
| q = [mp.mpf("0")] * N | |
| q[0] = mp.mpf("1") | |
| for _ in range(5): | |
| q = _conv_trunc(q, c, N) | |
| # r = p*q truncated | |
| r = _conv_trunc(p, q, N) | |
| # Prefactor for x*I0(5x)*K0(x)^5 after exponential cancellation: | |
| # x*I0(5x)*K0(x)^5 ~ c0 * sum_{k>=0} r_k / x^{2+k} | |
| c0 = mp.pi**2 / mp.sqrt(320) | |
| invX = mp.mpf(1) / X0 | |
| invXpow = invX # X0^-(k+1) | |
| s = mp.mpf("0") | |
| for k in range(N): | |
| s += r[k] * invXpow / (k + 1) | |
| invXpow *= invX | |
| return c0 * s | |
| def compute(): | |
| with mp.workdps(350): | |
| X0 = mp.mpf(200) | |
| def integrand(x): | |
| k0 = mp.besselk(0, x) | |
| return x * mp.besseli(0, 5 * x) * (k0**5) | |
| main = mp.quad( | |
| integrand, | |
| [mp.mpf("0"), mp.mpf("0.5"), 1, 2, 5, 10, 20, 40, 80, 120, 160, X0], | |
| ) | |
| tail = _tail_asymptotic(X0, N=300) | |
| res = main + tail | |
| return +res | |
| if __name__ == "__main__": | |
| print(str(compute())) |