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Merge pull request #1690 from edgarcosta/nospacebeforecolons
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removing spaces before colons
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@fredrik-johansson
fredrik-johansson authored Dec 31, 2023
2 parents 33999f4 + 4857b4e commit 622f031
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12 changes: 6 additions & 6 deletions doc/source/acb.rst
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Expand Up @@ -936,21 +936,21 @@ Gamma function

Computes the logarithmic sine function defined by

.. math ::
.. math::
S(z) = \log(\pi) - \log \Gamma(z) + \log \Gamma(1-z)
which is equal to

.. math ::
.. math::
S(z) = \int_{1/2}^z \pi \cot(\pi t) dt
where the path of integration goes through the upper half plane
if `0 < \arg(z) \le \pi` and through the lower half plane
if `-\pi < \arg(z) \le 0`. Equivalently,

.. math ::
.. math::
S(z) = \log(\sin(\pi(z-n))) \mp n \pi i, \quad n = \lfloor \operatorname{re}(z) \rfloor
Expand Down Expand Up @@ -983,7 +983,7 @@ Gamma function
The generalization to other values of *s* is due to
Espinosa and Moll [EM2004]_:

.. math ::
.. math::
\psi(s,z) = \frac{\zeta'(s+1,z) + (\gamma + \psi(-s)) \zeta(s+1,z)}{\Gamma(-s)}
Expand All @@ -997,7 +997,7 @@ Gamma function
in analogy with the logarithmic gamma function. The functional
equation

.. math ::
.. math::
\log G(z+1) = \log \Gamma(z) + \log G(z).
Expand All @@ -1007,7 +1007,7 @@ Gamma function
relation `G(z+1) = \Gamma(z) G(z)` together with the initial value
`G(1) = 1`. For general *z*, we use the formula

.. math ::
.. math::
\log G(z) = (z-1) \log \Gamma(z) - \zeta'(-1,z) + \zeta'(-1).
Expand Down
12 changes: 6 additions & 6 deletions doc/source/acb_calc.rst
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Expand Up @@ -104,7 +104,7 @@ Integration

Computes a rigorous enclosure of the integral

.. math ::
.. math::
I = \int_a^b f(t) dt
Expand Down Expand Up @@ -282,7 +282,7 @@ Local integration algorithms
For the interval `[-1,1]`, the error of the *n*-point Gauss-Legendre
rule is bounded by

.. math ::
.. math::
\left| I - \sum_{k=0}^{n-1} w_k f(x_k) \right| \le \frac{64 M}{15 (\rho-1) \rho^{2n-1}}
Expand Down Expand Up @@ -312,7 +312,7 @@ Integration (old)

Sets *bound* to a ball containing the value of the integral

.. math ::
.. math::
C(x,r) = \frac{1}{2 \pi r} \oint_{|z-x| = r} |f(z)| dz
= \int_0^1 |f(x+re^{2\pi i t})| dt
Expand All @@ -330,7 +330,7 @@ Integration (old)

Computes the integral

.. math ::
.. math::
I = \int_a^b f(t) dt
Expand All @@ -342,15 +342,15 @@ Integration (old)
formula. More precisely, if the Taylor series of *f* centered at the point
*m* is `f(m+x) = \sum_{n=0}^{\infty} a_n x^n`, then

.. math ::
.. math::
\int f(m+x) = \left( \sum_{n=0}^{N-1} a_n \frac{x^{n+1}}{n+1} \right)
+ \left( \sum_{n=N}^{\infty} a_n \frac{x^{n+1}}{n+1} \right).
For sufficiently small *x*, the second series converges and its
absolute value is bounded by

.. math ::
.. math::
\sum_{n=N}^{\infty} \frac{C(m,R)}{R^n} \frac{|x|^{n+1}}{N+1}
= \frac{C(m,R) R x}{(R-x)(N+1)} \left( \frac{x}{R} \right)^N.
Expand Down
24 changes: 12 additions & 12 deletions doc/source/acb_dirichlet.rst
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Expand Up @@ -7,7 +7,7 @@ This module allows working with values of Dirichlet characters,
Dirichlet L-functions, and related functions.
A Dirichlet L-function is the analytic continuation of an L-series

.. math ::
.. math::
L(s,\chi) = \sum_{k=1}^\infty \frac{\chi(k)}{k^s}
Expand Down Expand Up @@ -139,17 +139,17 @@ The Riemann-Siegel (RS) formula is implemented closely following
J. Arias de Reyna [Ari2011]_.
For `s = \sigma + it` with `t > 0`, the expansion takes the form

.. math ::
.. math::
\zeta(s) = \mathcal{R}(s) + X(s) \overline{\mathcal{R}}(1-s), \quad X(s) = \pi^{s-1/2} \frac{\Gamma((1-s)/2)}{\Gamma(s/2)}
where

.. math ::
.. math::
\mathcal{R}(s) = \sum_{k=1}^N \frac{1}{k^s} + (-1)^{N-1} U a^{-\sigma} \left[ \sum_{k=0}^K \frac{C_k(p)}{a^k} + RS_K \right]
.. math ::
.. math::
U = \exp\left(-i\left[ \frac{t}{2} \log\left(\frac{t}{2\pi}\right)-\frac{t}{2}-\frac{\pi}{8} \right]\right), \quad
a = \sqrt{\frac{t}{2\pi}}, \quad N = \lfloor a \rfloor, \quad p = 1-2(a-N).
Expand Down Expand Up @@ -276,7 +276,7 @@ Lerch transcendent

Computes the Lerch transcendent

.. math ::
.. math::
\Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(k+a)^s}
Expand All @@ -285,7 +285,7 @@ Lerch transcendent
The *direct* version evaluates a truncation of the defining series.
The *integral* version uses the Hankel contour integral

.. math ::
.. math::
\Phi(z,s,a) = -\frac{\Gamma(1-s)}{2 \pi i} \int_C \frac{(-t)^{s-1} e^{-a t}}{1 - z e^{-t}} dt
Expand All @@ -304,7 +304,7 @@ Stieltjes constants
`\gamma_n(a)` which is the coefficient in the Laurent series of the
Hurwitz zeta function at the pole

.. math ::
.. math::
\zeta(s,a) = \frac{1}{s-1} + \sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n(a) (s-1)^n.
Expand Down Expand Up @@ -547,14 +547,14 @@ Dirichlet L-functions
An error bound is computed via :func:`mag_hurwitz_zeta_uiui`.
If *s* is complex, replace it with its real part. Since

.. math ::
.. math::
\frac{1}{L(s,\chi)} = \prod_{p} \left(1 - \frac{\chi(p)}{p^s}\right)
= \sum_{k=1}^{\infty} \frac{\mu(k)\chi(k)}{k^s}
and the truncated product gives all smooth-index terms in the series, we have

.. math ::
.. math::
\left|\prod_{p < N} \left(1 - \frac{\chi(p)}{p^s}\right) - \frac{1}{L(s,\chi)}\right|
\le \sum_{k=N}^{\infty} \frac{1}{k^s} = \zeta(s,N).
Expand Down Expand Up @@ -593,7 +593,7 @@ Dirichlet L-functions
i.e. `L(s), L'(s), \ldots, L^{(len-1)}(s) / (len-1)!`.
If *deflate* is set, computes the expansion of

.. math ::
.. math::
L(s,\chi) - \frac{\sum_{k=1}^q \chi(k)}{(s-1)q}
Expand Down Expand Up @@ -623,7 +623,7 @@ Currently, these methods require *chi* to be a primitive character.
Computes the phase function used to construct the Z-function.
We have

.. math ::
.. math::
\theta(t) = -\frac{t}{2} \log(\pi/q) - \frac{i \log(\epsilon)}{2}
+ \frac{\log \Gamma((s+\delta)/2) - \log \Gamma((1-s+\delta)/2)}{2i}
Expand Down Expand Up @@ -777,7 +777,7 @@ and formulas described by David J. Platt in [Pla2017]_.

Compute `\Lambda(t) e^{\pi t/4}` where

.. math ::
.. math::
\Lambda(t) = \pi^{-\frac{it}{2}}
\Gamma\left(\frac{\frac{1}{2}+it}{2}\right)
Expand Down
34 changes: 17 additions & 17 deletions doc/source/acb_elliptic.rst
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Expand Up @@ -26,7 +26,7 @@ Complete elliptic integrals

Computes the complete elliptic integral of the first kind

.. math ::
.. math::
K(m) = \int_0^{\pi/2} \frac{dt}{\sqrt{1-m \sin^2 t}}
= \int_0^1
Expand All @@ -51,7 +51,7 @@ Complete elliptic integrals

Computes the complete elliptic integral of the second kind

.. math ::
.. math::
E(m) = \int_0^{\pi/2} \sqrt{1-m \sin^2 t} \, dt =
\int_0^1
Expand All @@ -64,7 +64,7 @@ Complete elliptic integrals

Evaluates the complete elliptic integral of the third kind

.. math ::
.. math::
\Pi(n, m) = \int_0^{\pi/2}
\frac{dt}{(1-n \sin^2 t) \sqrt{1-m \sin^2 t}} =
Expand All @@ -83,7 +83,7 @@ Legendre incomplete elliptic integrals
Evaluates the Legendre incomplete elliptic integral of the first kind,
given by

.. math ::
.. math::
F(\phi,m) = \int_0^{\phi} \frac{dt}{\sqrt{1-m \sin^2 t}}
= \int_0^{\sin \phi}
Expand All @@ -92,7 +92,7 @@ Legendre incomplete elliptic integrals
on the standard strip `-\pi/2 \le \operatorname{Re}(\phi) \le \pi/2`.
Outside this strip, the function extends quasiperiodically as

.. math ::
.. math::
F(\phi + n \pi, m) = 2 n K(m) + F(\phi,m), n \in \mathbb{Z}.
Expand All @@ -111,7 +111,7 @@ Legendre incomplete elliptic integrals
Evaluates the Legendre incomplete elliptic integral of the second kind,
given by

.. math ::
.. math::
E(\phi,m) = \int_0^{\phi} \sqrt{1-m \sin^2 t} \, dt =
\int_0^{\sin \phi}
Expand All @@ -120,7 +120,7 @@ Legendre incomplete elliptic integrals
on the standard strip `-\pi/2 \le \operatorname{Re}(\phi) \le \pi/2`.
Outside this strip, the function extends quasiperiodically as

.. math ::
.. math::
E(\phi + n \pi, m) = 2 n E(m) + E(\phi,m), n \in \mathbb{Z}.
Expand All @@ -139,7 +139,7 @@ Legendre incomplete elliptic integrals
Evaluates the Legendre incomplete elliptic integral of the third kind,
given by

.. math ::
.. math::
\Pi(n, \phi, m) = \int_0^{\phi}
\frac{dt}{(1-n \sin^2 t) \sqrt{1-m \sin^2 t}} =
Expand All @@ -149,7 +149,7 @@ Legendre incomplete elliptic integrals
on the standard strip `-\pi/2 \le \operatorname{Re}(\phi) \le \pi/2`.
Outside this strip, the function extends quasiperiodically as

.. math ::
.. math::
\Pi(n, \phi + k \pi, m) = 2 k \Pi(n,m) + \Pi(n,\phi,m), k \in \mathbb{Z}.
Expand Down Expand Up @@ -178,7 +178,7 @@ in [Car1995]_ and chapter 19 in [NIST2012]_.

Evaluates the Carlson symmetric elliptic integral of the first kind

.. math ::
.. math::
R_F(x,y,z) = \frac{1}{2}
\int_0^{\infty} \frac{dt}{\sqrt{(t+x)(t+y)(t+z)}}
Expand All @@ -205,7 +205,7 @@ in [Car1995]_ and chapter 19 in [NIST2012]_.

Evaluates the Carlson symmetric elliptic integral of the second kind

.. math ::
.. math::
R_G(x,y,z) = \frac{1}{4} \int_0^{\infty}
\frac{t}{\sqrt{(t+x)(t+y)(t+z)}}
Expand All @@ -224,7 +224,7 @@ in [Car1995]_ and chapter 19 in [NIST2012]_.

Evaluates the Carlson symmetric elliptic integral of the third kind

.. math ::
.. math::
R_J(x,y,z,p) = \frac{3}{2}
\int_0^{\infty} \frac{dt}{(t+p)\sqrt{(t+x)(t+y)(t+z)}}
Expand Down Expand Up @@ -285,15 +285,15 @@ The main reference is chapter 23 in [NIST2012]_.

Computes Weierstrass's elliptic function

.. math ::
.. math::
\wp(z, \tau) = \frac{1}{z^2} + \sum_{n^2+m^2 \ne 0}
\left[ \frac{1}{(z+m+n\tau)^2} - \frac{1}{(m+n\tau)^2} \right]
which satisfies `\wp(z, \tau) = \wp(z + 1, \tau) = \wp(z + \tau, \tau)`.
To evaluate the function efficiently, we use the formula

.. math ::
.. math::
\wp(z, \tau) = \pi^2 \theta_2^2(0,\tau) \theta_3^2(0,\tau)
\frac{\theta_4^2(z,\tau)}{\theta_1^2(z,\tau)} -
Expand Down Expand Up @@ -337,7 +337,7 @@ The main reference is chapter 23 in [NIST2012]_.
satisfies `\wp(\wp^{-1}(z, \tau), \tau) = z`. This function is given
by the elliptic integral

.. math ::
.. math::
\wp^{-1}(z, \tau) = \frac{1}{2} \int_z^{\infty} \frac{dt}{\sqrt{(t-e_1)(t-e_2)(t-e_3)}}
= R_F(z-e_1,z-e_2,z-e_3).
Expand All @@ -346,7 +346,7 @@ The main reference is chapter 23 in [NIST2012]_.

Computes the Weierstrass zeta function

.. math ::
.. math::
\zeta(z, \tau) = \frac{1}{z} + \sum_{n^2+m^2 \ne 0}
\left[ \frac{1}{z-m-n\tau} + \frac{1}{m+n\tau} + \frac{z}{(m+n\tau)^2} \right]
Expand All @@ -358,7 +358,7 @@ The main reference is chapter 23 in [NIST2012]_.

Computes the Weierstrass sigma function

.. math ::
.. math::
\sigma(z, \tau) = z \prod_{n^2+m^2 \ne 0}
\left[ \left(1-\frac{z}{m+n\tau}\right)
Expand Down
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