7^(h^ d Z ddlmZ ddlmZ ddlmZmZmZ ddl m
Z
ddl m Z m
Z
mZ ddlmZ ddlmZ dd lmZ dd
lmZ dd lmZmZmZmZ dd l m Z m Z m Z m Z dd
l!m"Z"m#Z#m$Z$ ddl%m&Z&m'Z' ddl(m)Z) ddl*m+Z+ ddl,m-Z- G d de Z. G d de Z/ G d de Z0 G d de Z1 G d de Z2 G d d e Z3ed Z4y ) z$ Riemann zeta and related function. )Add)cacheit)ArgumentIndexError
expand_mulDefinedFunction) fuzzy_not)piIInteger)Eq)S)Dummy)sympify) bernoulli factorialgenocchiharmonic)re
unpolarifyAbs
polar_lift)log exp_polarexp)ceilingfloor)sqrt) Piecewise)Polyc 0 e Zd ZdZd ZddZd Zd Zd Zy) lerchphia^
Lerch transcendent (Lerch phi function).
Explanation
===========
For $\operatorname{Re}(a) > 0$, $|z| < 1$ and $s \in \mathbb{C}$, the
Lerch transcendent is defined as
.. math :: \Phi(z, s, a) = \sum_{n=0}^\infty \frac{z^n}{(n + a)^s},
where the standard branch of the argument is used for $n + a$,
and by analytic continuation for other values of the parameters.
A commonly used related function is the Lerch zeta function, defined by
.. math:: L(q, s, a) = \Phi(e^{2\pi i q}, s, a).
**Analytic Continuation and Branching Behavior**
It can be shown that
.. math:: \Phi(z, s, a) = z\Phi(z, s, a+1) + a^{-s}.
This provides the analytic continuation to $\operatorname{Re}(a) \le 0$.
Assume now $\operatorname{Re}(a) > 0$. The integral representation
.. math:: \Phi_0(z, s, a) = \int_0^\infty \frac{t^{s-1} e^{-at}}{1 - ze^{-t}}
\frac{\mathrm{d}t}{\Gamma(s)}
provides an analytic continuation to $\mathbb{C} - [1, \infty)$.
Finally, for $x \in (1, \infty)$ we find
.. math:: \lim_{\epsilon \to 0^+} \Phi_0(x + i\epsilon, s, a)
-\lim_{\epsilon \to 0^+} \Phi_0(x - i\epsilon, s, a)
= \frac{2\pi i \log^{s-1}{x}}{x^a \Gamma(s)},
using the standard branch for both $\log{x}$ and
$\log{\log{x}}$ (a branch of $\log{\log{x}}$ is needed to
evaluate $\log{x}^{s-1}$).
This concludes the analytic continuation. The Lerch transcendent is thus
branched at $z \in \{0, 1, \infty\}$ and
$a \in \mathbb{Z}_{\le 0}$. For fixed $z, a$ outside these
branch points, it is an entire function of $s$.
Examples
========
The Lerch transcendent is a fairly general function, for this reason it does
not automatically evaluate to simpler functions. Use ``expand_func()`` to
achieve this.
If $z=1$, the Lerch transcendent reduces to the Hurwitz zeta function:
>>> from sympy import lerchphi, expand_func
>>> from sympy.abc import z, s, a
>>> expand_func(lerchphi(1, s, a))
zeta(s, a)
More generally, if $z$ is a root of unity, the Lerch transcendent
reduces to a sum of Hurwitz zeta functions:
>>> expand_func(lerchphi(-1, s, a))
zeta(s, a/2)/2**s - zeta(s, a/2 + 1/2)/2**s
If $a=1$, the Lerch transcendent reduces to the polylogarithm:
>>> expand_func(lerchphi(z, s, 1))
polylog(s, z)/z
More generally, if $a$ is rational, the Lerch transcendent reduces
to a sum of polylogarithms:
>>> from sympy import S
>>> expand_func(lerchphi(z, s, S(1)/2))
2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -
polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))
>>> expand_func(lerchphi(z, s, S(3)/2))
-2**s/z + 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -
polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))/z
The derivatives with respect to $z$ and $a$ can be computed in
closed form:
>>> lerchphi(z, s, a).diff(z)
(-a*lerchphi(z, s, a) + lerchphi(z, s - 1, a))/z
>>> lerchphi(z, s, a).diff(a)
-s*lerchphi(z, s + 1, a)
See Also
========
polylog, zeta
References
==========
.. [1] Bateman, H.; Erdelyi, A. (1953), Higher Transcendental Functions,
Vol. I, New York: McGraw-Hill. Section 1.11.
.. [2] https://dlmf.nist.gov/25.14
.. [3] https://en.wikipedia.org/wiki/Lerch_transcendent
c @ | j \ }}}|dk( r t || S |j r|dk rt d }t ||z | z | }dd|z
z }t
j }t |j D ] } || |z z
}||j | z } |j || S |j rt
j }
t
j } |dkD rPt | } | |k( r| dz } || z }|| z } t t | D
cg c] }
||
| z
z ||
z |z z c}
}
nR|dk rMt | dz } || z
}|| z } t t | D
cg c] }
|| dz
|
z
z ||
z
dz
|z z c}
}
t |j |j" g \ }} t% dt& z t( z | z }|d| z z }t+ | }g }t | D ]S }
t- |||
z |z }t/ |t, r |j0 di |}|j3 |||
z |z |z z U |
| | |dz
z z t | z z S t/ |t4 r'|j d t& t( z z j s|dt( t( fv r|dk( rt ddg \ }}ny|t( k( rt ddg \ }}n_|t( k( rt ddg \ }}nD|j d dt& z t( z z }t |j |j" g \ }}t t | D
cg c]= }
t5 dt& z t( z |
z |z |z ||z z t ||
|z |z z ? c}
S t7 ||| S c c}
w c c}
w c c}
w )N r t ) argszeta
is_Integerr r r
Zeroreversed
all_coeffsdiffsubs is_RationalOner r rangepqr r r
r polylog
isinstance_eval_expand_funcappendr r! )selfhintszsar$ r4 startrescaddmulnkmzetrootup_zetaddargsr5 args d/var/www/skyplay_api_hub/venv/lib/python3.12/site-packages/sympy/functions/special/zeta_functions.pyr8 zlerchphi._eval_expand_func s ))1a 61: < ! == &&C%%C1u!H6FAQ1"gE!HEqQQZKQ
2EFa1"IMQd58LaAA NAEAI> 9LMacc133Z=DAqAbDF1H%Cqs8D _FG1X
6AsAvd{+ a)+++4e4Aq&!)D.1!4 45
6
QQZ W
55 5 a 166!9bd#3"@"@A"aRSQSDTBw!Qy1a!Qy1qb"az1ffQi2a(#%% (1"'(, QrT!VAXaZ\*1a4/QQ 0BB , -
- 1a E F
M4,s N' N>ANc | j \ }}}|dk( r| t ||dz | z S |dk( r$t ||dz
| |t ||| z z
|z S t )N r# )r) r! r )r: argindexr< r= r> s rL fdiffzlerchphi.fdiff sj ))1a q=2hq!a%++ +
] QAq)Ahq!Q.?,??B B$ $ c L | j }|j | r|S | S N)r8 has)r: targetr@ s rL _eval_rewrite_helperz lerchphi._eval_rewrite_helper s% $$& 776?JKrQ c , | j t S rS )rV r* r: r< r= r> kwargss rL _eval_rewrite_as_zetaz lerchphi._eval_rewrite_as_zeta s ((..rQ c , | j t S rS )rV r6 rX s rL _eval_rewrite_as_polylogz!lerchphi._eval_rewrite_as_polylog s ((11rQ Nr# ) __name__
__module__ __qualname____doc__r8 rP rV rZ r\ r( rQ rL r! r! s# gR?!B%/2rQ r! c N e Zd ZdZed ZddZd Zd Zd Z d fd Z
xZ S )
r6 a
Polylogarithm function.
Explanation
===========
For $|z| < 1$ and $s \in \mathbb{C}$, the polylogarithm is
defined by
.. math:: \operatorname{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s},
where the standard branch of the argument is used for $n$. It admits
an analytic continuation which is branched at $z=1$ (notably not on the
sheet of initial definition), $z=0$ and $z=\infty$.
The name polylogarithm comes from the fact that for $s=1$, the
polylogarithm is related to the ordinary logarithm (see examples), and that
.. math:: \operatorname{Li}_{s+1}(z) =
\int_0^z \frac{\operatorname{Li}_s(t)}{t} \mathrm{d}t.
The polylogarithm is a special case of the Lerch transcendent:
.. math:: \operatorname{Li}_{s}(z) = z \Phi(z, s, 1).
Examples
========
For $z \in \{0, 1, -1\}$, the polylogarithm is automatically expressed
using other functions:
>>> from sympy import polylog
>>> from sympy.abc import s
>>> polylog(s, 0)
0
>>> polylog(s, 1)
zeta(s)
>>> polylog(s, -1)
-dirichlet_eta(s)
If $s$ is a negative integer, $0$ or $1$, the polylogarithm can be
expressed using elementary functions. This can be done using
``expand_func()``:
>>> from sympy import expand_func
>>> from sympy.abc import z
>>> expand_func(polylog(1, z))
-log(1 - z)
>>> expand_func(polylog(0, z))
z/(1 - z)
The derivative with respect to $z$ can be computed in closed form:
>>> polylog(s, z).diff(z)
polylog(s - 1, z)/z
The polylogarithm can be expressed in terms of the lerch transcendent:
>>> from sympy import lerchphi
>>> polylog(s, z).rewrite(lerchphi)
z*lerchphi(z, s, 1)
See Also
========
zeta, lerchphi
c |j ru|t j u r t | S |t j u r t | S |t j u rt j S |dk( rt }||v r|| S |j rt j S |j t j }|r t | S |du rK|t j u r|d|z
z S |t j u r |d|z
dz z S |j r|d|z
z S |j t t r4|s t | t j k dk( r | |t | S y y )Nr% Fr# T) is_numberr
r2 r* NegativeOne
dirichlet_etar, _dilogtableis_zeroequalsrT r r r r )clsr= r<
dilogtablezones rL evalz polylog.eval% s- ;;AEEzAwamm#%a(((affvv
a (]
?%a=( 9966M xx 7N
U]
AFF{!a%y amm#!a%!|#yy!a%y
55J 'Tc!fo$5Nq*Q-( ( 6O 'rQ c Z | j \ }}|dk( rt |dz
| |z S t )Nr% r# )r) r6 r )r: rO r= r< s rL rP z
polylog.fdiffL s2 yy1 q=1q5!$Q& & rQ c " |t ||d z S Nr# r! )r: r= r< rY s rL _eval_rewrite_as_lerchphiz!polylog._eval_rewrite_as_lerchphiR s !Q"""rQ c ( | j \ }}|dk( rt d|z
S |j rX|dk rSt d }|d|z
z }t | D ] }||j | z } t
| j || S t || S )Nr# r u) r) r r+ r r3 r/ r r0 r6 )r: r; r= r< rt r? _s rL r8 zpolylog._eval_expand_funcU s yy1 6AJ; <*>3CHB
::
A 3 qu~!Q$NOOAq$5==?; HF q$ )AAIDHHT!R%Z() Aw{"Wd1!QdCC# 34
s 3D6 6E
E
r] )r ) r^ r_ r` ra classmethodrm rP rr r8 rw r
__classcell__r s @rL r6 r6 s? CJ $) $)L! #
D DrQ r6 c ` e Zd ZdZed
d Zd dZd dZd dZd Z d Z
d dZ fd Z xZ
S ) r* a
Hurwitz zeta function (or Riemann zeta function).
Explanation
===========
For $\operatorname{Re}(a) > 0$ and $\operatorname{Re}(s) > 1$, this
function is defined as
.. math:: \zeta(s, a) = \sum_{n=0}^\infty \frac{1}{(n + a)^s},
where the standard choice of argument for $n + a$ is used. For fixed
$a$ not a nonpositive integer the Hurwitz zeta function admits a
meromorphic continuation to all of $\mathbb{C}$; it is an unbranched
function with a simple pole at $s = 1$.
The Hurwitz zeta function is a special case of the Lerch transcendent:
.. math:: \zeta(s, a) = \Phi(1, s, a).
This formula defines an analytic continuation for all possible values of
$s$ and $a$ (also $\operatorname{Re}(a) < 0$), see the documentation of
:class:`lerchphi` for a description of the branching behavior.
If no value is passed for $a$ a default value of $a = 1$ is assumed,
yielding the Riemann zeta function.
Examples
========
For $a = 1$ the Hurwitz zeta function reduces to the famous Riemann
zeta function:
.. math:: \zeta(s, 1) = \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.
>>> from sympy import zeta
>>> from sympy.abc import s
>>> zeta(s, 1)
zeta(s)
>>> zeta(s)
zeta(s)
The Riemann zeta function can also be expressed using the Dirichlet eta
function:
>>> from sympy import dirichlet_eta
>>> zeta(s).rewrite(dirichlet_eta)
dirichlet_eta(s)/(1 - 2**(1 - s))
The Riemann zeta function at nonnegative even and negative integer
values is related to the Bernoulli numbers and polynomials:
>>> zeta(2)
pi**2/6
>>> zeta(4)
pi**4/90
>>> zeta(0)
-1/2
>>> zeta(-1)
-1/12
>>> zeta(-4)
0
The specific formulae are:
.. math:: \zeta(2n) = -\frac{(2\pi i)^{2n} B_{2n}}{2(2n)!}
.. math:: \zeta(-n,a) = -\frac{B_{n+1}(a)}{n+1}
No closed-form expressions are known at positive odd integers, but
numerical evaluation is possible:
>>> zeta(3).n()
1.20205690315959
The derivative of $\zeta(s, a)$ with respect to $a$ can be computed:
>>> from sympy.abc import a
>>> zeta(s, a).diff(a)
-s*zeta(s + 1, a)
However the derivative with respect to $s$ has no useful closed form
expression:
>>> zeta(s, a).diff(s)
Derivative(zeta(s, a), s)
The Hurwitz zeta function can be expressed in terms of the Lerch
transcendent, :class:`~.lerchphi`:
>>> from sympy import lerchphi
>>> zeta(s, a).rewrite(lerchphi)
lerchphi(1, s, a)
See Also
========
dirichlet_eta, lerchphi, polylog
References
==========
.. [1] https://dlmf.nist.gov/25.11
.. [2] https://en.wikipedia.org/wiki/Hurwitz_zeta_function
c |t j u r | | S |t j u s|t j u rt j S |t j u rt j S |t j u rt j S |t j u rt j
S |j }|t j }|r!|j rt d|z
| |dz
z S |t j u r?|r<|j r/dt z t z |z t | z dt | z z S y y |r0|j r$|j r | | t |dz
| z
S |j r:|j r-|j du s|j du rt j S y y y )Nr# r% F)r
r2 r~ ComplexInfinityInfinityr, r+ is_nonpositiver is_evenr r
r r r
is_integer)rj r= r> sints rL rm z zeta.eval s[ :q6M
!%%Z1:55L
!%%Z$$ $
!**_55L
!**_66M|| 9A A$$ QqS!$!, ,
!%%Z 2a! |il2a ! nEE "t
allq}}q6HQqS! ,, ,
\\a . . &!*:*:e*C55L +D /\rQ c |dk( rS|j rG|j r;|j r/dt z t z |z t | z dt
| z z S t d|z
| |dz
z S )Nr# r% )r is_nonnegativer r r
r r r: r= r> rY s rL _eval_rewrite_as_bernoulliz zeta._eval_rewrite_as_bernoulli sa 6allq'7'7AIIrT!VaK<)A,.!IaL.A A1a AaC((rQ c \ |dk7 r| S | j d }t | ddd|z
z z
z S )Nr# r r% )r) rf r s rL _eval_rewrite_as_dirichlet_etaz#zeta._eval_rewrite_as_dirichlet_eta s7 6K IIaL Q QQZ 00rQ c t d|| S rp rq r s rL rr z zeta._eval_rewrite_as_lerchphi s 1a rQ c L t | j d dz
j S )Nr r# )r r) rh )r: s rL _eval_is_finitezzeta._eval_is_finite s $))A,*3344rQ c h | j d }t | j dkD r| j d nt j }|j r_|j
rt
| t |dz
| z
S |j r,|j du s|j du rt j S | S )Nr r# F)
r) lenr
r2 r r r* r r r~ )r: r; r= r> s rL r8 zzeta._eval_expand_func" s IIaL NQ .DIIaLAEE <<}}Aw!A#q!111 Q\\U%:$$-uu rQ c t | j dk( r| j \ }}n| j dz \ }}|dk( r| t |dz | z S t )Nr% r] r# )r r) r* r )r: rO r= r> s rL rP z
zeta.fdiff- sU tyy>Q 99DAq99t#DAq q=2d1q5!n$ $$ $rQ c J t | j dk( r| j \ }}n!| j t j fz \ }} |j | \ }}|j r|j st
t t | + ||| S # t
$ r | cY S w xY w)Nr% )r r ) r r) r
r2 r r r r r r* _eval_as_leading_term) r: r r r r= r> rA er s rL r zzeta._eval_as_leading_term7 s tyy>Q 99DAq99x'DAq ::a=DAq
== % %T46qt$6OO
# K s B B"!B"rS r] )r^ r_ r` ra r rm r r rr r r8 rP r r r s @rL r* r* sH hT 4)
1 !5 %P PrQ r* c N e Zd ZdZedd ZddZej fdZ d Z
y) rf a
Dirichlet eta function.
Explanation
===========
For $\operatorname{Re}(s) > 0$ and $0 < x \le 1$, this function is defined as
.. math:: \eta(s, a) = \sum_{n=0}^\infty \frac{(-1)^n}{(n+a)^s}.
It admits a unique analytic continuation to all of $\mathbb{C}$ for any
fixed $a$ not a nonpositive integer. It is an entire, unbranched function.
It can be expressed using the Hurwitz zeta function as
.. math:: \eta(s, a) = \zeta(s,a) - 2^{1-s} \zeta\left(s, \frac{a+1}{2}\right)
and using the generalized Genocchi function as
.. math:: \eta(s, a) = \frac{G(1-s, a)}{2(s-1)}.
In both cases the limiting value of $\log2 - \psi(a) + \psi\left(\frac{a+1}{2}\right)$
is used when $s = 1$.
Examples
========
>>> from sympy import dirichlet_eta, zeta
>>> from sympy.abc import s
>>> dirichlet_eta(s).rewrite(zeta)
Piecewise((log(2), Eq(s, 1)), ((1 - 2**(1 - s))*zeta(s), True))
See Also
========
zeta
References
==========
.. [1] https://en.wikipedia.org/wiki/Dirichlet_eta_function
.. [2] Peter Luschny, "An introduction to the Bernoulli function",
https://arxiv.org/abs/2009.06743
Nc |t j u r | | S |?|dk( r t d S t | }|j t sddd|z
z z
|z S y |dk( r)ddlm} t d || z
||dz dz z S t || }t ||dz dz }|j t s$|j t s|dd|z
z |z z
S y y )Nr# r% r digamma)r
r2 r r* rT 'sympy.functions.special.gamma_functionsr )rj r= r> r< r z1z2s rL rm zdirichlet_eta.evalw s :q6M 9Av1v
QA55;A!H ))
!V Gq6GAJ&!A#q)99 9
!QZ
!ac1W
vvd|BFF4LAaC2
% % %1|rQ c
J ddl m} |dk( r8t t d t |d fddd|z
z z
t | z df S t t d || z
||dz dz z t |d ft || dd|z
z t ||dz dz z z
df S )Nr r r# r% T)r r r r r r* r: r= r> rY r s rL rZ z#dirichlet_eta._eval_rewrite_as_zeta s C 6 c!fbAh /1q1Q3x<472JD1QR R#a&71:-1a0@@"Q(Kaa!A#ha!A#q)9 994@B BrQ c ddl m} t t d || z
||dz dz z t |d ft d|z
| d|dz
z z df S )Nr r r% r# T)r r r r r r r s rL _eval_rewrite_as_genocchiz'dirichlet_eta._eval_rewrite_as_genocchi s^ C#a&71:-1a0@@"Q(K!A#q!Q!A#Y/68 8rQ c t d | j D r$| j t j | S y )Nc 3 4 K | ] }|j y wrS )rd ).0is rL z,dirichlet_eta._eval_evalf.. s .qq{{.s )allr) rewriter* _eval_evalf)r: precs rL r zdirichlet_eta._eval_evalf s3 .DII. .<<%11$7 7 /rQ rS r] ) r^ r_ r` ra r rm rZ r
r2 r r r( rQ rL rf rf H s5 ,\ & &$B ./UU 8
8rQ rf c & e Zd ZdZed Zd Zy)
riemann_xia
Riemann Xi function.
Examples
========
The Riemann Xi function is closely related to the Riemann zeta function.
The zeros of Riemann Xi function are precisely the non-trivial zeros
of the zeta function.
>>> from sympy import riemann_xi, zeta
>>> from sympy.abc import s
>>> riemann_xi(s).rewrite(zeta)
s*(s - 1)*gamma(s/2)*zeta(s)/(2*pi**(s/2))
References
==========
.. [1] https://en.wikipedia.org/wiki/Riemann_Xi_function
c ddl m} t | }|t j t j
fv rt j S t |t s'||dz
z ||dz z |z dt |dz z z z S y Nr )gammar# r% ) r r r* r
r, r2 Halfr7 r )rj r= r r< s rL rm zriemann_xi.eval sk A G 66M!T"a!e9U1Q3Z')1R!A#Y;7 7 #rQ c n ddl m} ||dz
z ||dz z t | z dt |dz z z z S r )r r r* r )r: r= rY r s rL rZ z riemann_xi._eval_rewrite_as_zeta s: A!a%yqs#DG+QrAaCy[99rQ N)r^ r_ r` ra r rm rZ r( rQ rL r r s . 8 8:rQ r c " e Zd ZdZedd Zy) stieltjesa
Represents Stieltjes constants, $\gamma_{k}$ that occur in
Laurent Series expansion of the Riemann zeta function.
Examples
========
>>> from sympy import stieltjes
>>> from sympy.abc import n, m
>>> stieltjes(n)
stieltjes(n)
The zero'th stieltjes constant:
>>> stieltjes(0)
EulerGamma
>>> stieltjes(0, 1)
EulerGamma
For generalized stieltjes constants:
>>> stieltjes(n, m)
stieltjes(n, m)
Constants are only defined for integers >= 0:
>>> stieltjes(-1)
zoo
References
==========
.. [1] https://en.wikipedia.org/wiki/Stieltjes_constants
Nc r |Ut | }|t j u rt j S |j r |j rt j
S |j ry|t j u rt j S |dk rt j
S |j st j
S |t j u r|dv rt j S |j rt j
S |j r|dv rt j S |j dk( rt j
S y )Nr rp F) r r
r~ r+ r r is_Numberr,
EulerGammais_extended_negativerh r )rj rD r> s rL rm zstieltjes.eval s =
AAEEzuu || 0 0((( ;;AEEzuu Q(((\\(((affi ||# ! !$$ $ 99i << <<5 $$ $ !rQ rS )r^ r_ r` ra r rm r( rQ rL r r s "H % %rQ r c t j t dz dz t d dz dz z
t d t dz dz t
t z t d z z
t
d dz
dz t dz dz t t
d dz
dz dz dz z t
d dz dz t dz dz t t
d dz dz dz z
dt
d z
dz t dz dz t t
d dz
dz dz z
t
d dz
dz t dz dz t t
d dz
dz dz z
t
t
t j z t dz d z z
t
t
t j z t dz d z z
dt
z
t dz d
z t
t j z z
t t
z dz t d z z
dt
z t dz d
z t
t j z z t t
z dz t d z z dt
z
dz t d dz d z t t
z t d z d z z dt dz z d z z t
t j z z
i S )
Nr% r' r#
rN 0 ` )r
r r r r r
r Catalanr( rQ rL rg rg s
Ab3q619Q;&
RU1WqtCF{*
q'A+qBE6"9sDGAIq='91'>
q'A+qBE6"9sDGAIq='91'<<
T!Wa"a%(S$q'!)Q %7%::
a1a"a%(S$q'!)Q %7%:: AaiiK"a%( "
aR \BE"H
$ AAb1QYY;&AaA 6 AAb1QYY;&AaA 6
Q SVQYJqL2a4A;q=01RU72:=!)) K rQ N)5ra sympy.core.addr sympy.core.cacher sympy.core.functionr r r sympy.core.logicr sympy.core.numbersr r
r sympy.core.relationalr sympy.core.singletonr
sympy.core.symbolr sympy.core.sympifyr %sympy.functions.combinatorial.numbersr r r r $sympy.functions.elementary.complexesr r r r &sympy.functions.elementary.exponentialr r r #sympy.functions.elementary.integersr r (sympy.functions.elementary.miscellaneousr $sympy.functions.elementary.piecewiser sympy.polys.polytoolsr r! r6 r* rf r r rg r( rQ rL r s * $ O O & - - $ " # & Z Z P P F F > 9 : &2 2LeDo eDXzP? zPzP8O P8f$: $:N?% ?%D
rQ