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https://github.com/vale981/arb
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177 lines
5.6 KiB
ReStructuredText
177 lines
5.6 KiB
ReStructuredText
.. _algorithms_hypergeometric:
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Algorithms for hypergeometric functions
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===============================================================================
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.. _algorithms_hypergeometric_asymptotic_confluent:
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Asymptotic series for the confluent hypergeometric function
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-------------------------------------------------------------------------------
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Let `U(a,b,z)` denote the confluent hypergeometric function of the second
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kind with the principal branch cut, and
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let `U^{*} = z^a U(a,b,z)`.
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For all `z \ne 0` and `b \notin \mathbb{Z}` (but valid for all `b` as a limit),
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we have (DLMF 13.2.42)
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.. math ::
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U(a,b,z)
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= \frac{\Gamma(1-b)}{\Gamma(a-b+1)} M(a,b,z)
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+ \frac{\Gamma(b-1)}{\Gamma(a)} z^{1-b} M(a-b+1,2-b,z).
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Moreover, for all `z \ne 0` we have
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.. math ::
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\frac{{}_1F_1(a,b,z)}{\Gamma(b)}
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= \frac{(-z)^{-a}}{\Gamma(b-a)} U^{*}(a,b,z)
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+ \frac{z^{a-b} e^z}{\Gamma(a)} U^{*}(b-a,b,-z)
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which is equivalent to DLMF 13.2.41 (but simpler in form).
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We have the asymptotic expansion
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.. math ::
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U^{*}(a,b,z) \sim {}_2F_0(a, a-b+1, -1/z)
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where `{}_2F_0(a,b,z)` denotes a formal hypergeometric series, i.e.
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.. math ::
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U^{*}(a,b,z) = \sum_{k=0}^{n-1} \frac{(a)_k (a-b+1)_k}{k! (-z)^k} + \varepsilon_n(z).
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The error term `\varepsilon_n(z)` is bounded according to DLMF 13.7.
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A case distinction is made depending on whether `z` lies in one
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of three regions which we index by `R`.
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Our formula for the error bound increases with the value of `R`, so we
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can always choose the larger out of two indices if `z` lies in
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the union of two regions.
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Let `r = |b-2a|`.
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If `\operatorname{Re}(z) \ge r`, set `R = 1`.
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Otherwise, if `\operatorname{Im}(z) \ge r` or `\operatorname{Re}(z) \ge 0 \land |z| \ge r`, set `R = 2`.
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Otherwise, if `|z| \ge 2r`, set `R = 3`.
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Otherwise, the bound is infinite.
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If the bound is finite, we have
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.. math ::
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|\varepsilon_n(z)| \le 2 \alpha C_n \left|\frac{(a)_n (a-b+1)_n}{n! z^n} \right| \exp(2 \alpha \rho C_1 / |z|)
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in terms of the following auxiliary quantities
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.. math ::
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\sigma = |(b-2a)/z|
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C_n = \begin{cases}
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1 & \text{if } R = 1 \\
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\chi(n) & \text{if } R = 2 \\
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(\chi(n) + \sigma \nu^2 n) \nu^n & \text{if } R = 3
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\end{cases}
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\nu = \left(\tfrac{1}{2} + \tfrac{1}{2}\sqrt{1-4\sigma^2}\right)^{-1/2} \le 1 + 2 \sigma^2
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\chi(n) = \sqrt{\pi} \Gamma(\tfrac{1}{2}n+1) / \Gamma(\tfrac{1}{2} n + \tfrac{1}{2})
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\sigma' = \begin{cases}
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\sigma & \text{if } R \ne 3 \\
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\nu \sigma & \text{if } R = 3
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\end{cases}
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\alpha = (1 - \sigma')^{-1}
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\rho = \tfrac{1}{2} |2a^2-2ab+b| + \sigma' (1+ \tfrac{1}{4} \sigma') (1-\sigma')^{-2}
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.. _algorithms_hypergeometric_asymptotic_airy:
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Asymptotic series for Airy functions
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-------------------------------------------------------------------------------
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Error bounds are based on Olver (DLMF section 9.7).
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For `\arg(z) < \pi` and `\zeta = (2/3) z^{3/2}`, we have
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.. math ::
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\operatorname{Ai}(z) = \frac{e^{-\zeta}}{2 \sqrt{\pi} z^{1/4}} \left[S_n(\zeta) + R_n(z)\right], \quad
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\operatorname{Ai}'(z) = -\frac{z^{1/4} e^{-\zeta}}{2 \sqrt{\pi}} \left[(S'_n(\zeta) + R'_n(z)\right]
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S_n(\zeta) = \sum_{k=0}^{n-1} (-1)^k \frac{u(k)}{\zeta^k}, \quad
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S'_n(\zeta) = \sum_{k=0}^{n-1} (-1)^k \frac{v(k)}{\zeta^k}
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u(k) = \frac{(1/6)_k (5/6)_k}{2^k k!}, \quad
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v(k) = \frac{6k+1}{1-6k} u(k).
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Assuming that *n* is positive, the error terms are bounded by
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.. math ::
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|R_n(z)| \le C |u(n)| |\zeta|^{-n}, \quad |R'_n(z)| \le C |v(n)| |\zeta|^{-n}
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where
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.. math ::
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C = \begin{cases}
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2 \exp(7 / (36 |\zeta|)) & |\arg(z)| \le \pi/3 \\
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2 \chi(n) \exp(7 \pi / (72 |\zeta|)) & \pi/3 \le |\arg(z)| \le 2\pi/3 \\
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4 \chi(n) \exp(7 \pi / (36 |\operatorname{re}(\zeta)|)) |\cos(\arg(\zeta))|^{-n} & 2\pi/3 \le |\arg(z)| < \pi.
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\end{cases}
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For computing Bi when *z* is roughly in the positive half-plane, we use the
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connection formulas
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.. math ::
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\operatorname{Bi}(z) = -i (2 w^{+1} \operatorname{Ai}(z w^{-2}) - \operatorname{Ai}(z))
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\operatorname{Bi}(z) = +i (2 w^{-1} \operatorname{Ai}(z w^{+2}) - \operatorname{Ai}(z))
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where `w = \exp(\pi i/3)`. Combining roots of unity gives
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.. math ::
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\operatorname{Bi}(z) = \frac{1}{2 \sqrt{\pi} z^{1/4}} [2X + iY]
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\operatorname{Bi}(z) = \frac{1}{2 \sqrt{\pi} z^{1/4}} [2X - iY]
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X = \exp(+\zeta) [S_n(-\zeta) + R_n(z w^{\mp 2})], \quad Y = \exp(-\zeta) [S_n(\zeta) + R_n(z)]
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where the upper formula is valid
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for `-\pi/3 < \arg(z) < \pi` and the lower formula is valid for `-\pi < \arg(z) < \pi/3`.
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We proceed analogously for the derivative of Bi.
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In the negative half-plane, we use the connection formulas
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.. math ::
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\operatorname{Ai}(z) = e^{+\pi i/3} \operatorname{Ai}(z_1) + e^{-\pi i/3} \operatorname{Ai}(z_2)
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\operatorname{Bi}(z) = e^{-\pi i/6} \operatorname{Ai}(z_1) + e^{+\pi i/6} \operatorname{Ai}(z_2)
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where `z_1 = -z e^{+\pi i/3}`, `z_2 = -z e^{-\pi i/3}`.
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Provided that `|\arg(-z)| < 2 \pi / 3`, we have
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`|\arg(z_1)|, |\arg(z_2)| < \pi`, and thus the asymptotic expansion
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for Ai can be used. As before, we collect roots of unity to obtain
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.. math ::
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\operatorname{Ai}(z) = A_1 [S_n(i \zeta) + R_n(z_1)]
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+ A_2 [S_n(-i \zeta) + R_n(z_2)]
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\operatorname{Bi}(z) = A_3 [S_n(i \zeta) + R_n(z_1)]
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+ A_4 [S_n(-i \zeta) + R_n(z_2)]
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where `\zeta = (2/3) (-z)^{3/2}` and
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.. math ::
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A_1 = \frac{\exp(-i (\zeta - \pi/4))}{2 \sqrt{\pi} (-z)^{1/4}}, \quad
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A_2 = \frac{\exp(+i (\zeta - \pi/4))}{2 \sqrt{\pi} (-z)^{1/4}}, \quad
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A_3 = -i A_1, \quad
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A_4 = +i A_2.
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The differentiated formulas are analogous.
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