"""Laplace Transforms"""
from sympy.core import S, pi, I
from sympy.core.add import Add
from sympy.core.cache import cacheit
from sympy.core.function import (
AppliedUndef, Derivative, expand, expand_complex, expand_mul, expand_trig,
Lambda, WildFunction, diff)
from sympy.core.mul import Mul, prod
from sympy.core.relational import _canonical, Ge, Gt, Lt, Unequality, Eq
from sympy.core.sorting import ordered
from sympy.core.symbol import Dummy, symbols, Wild
from sympy.functions.elementary.complexes import (
re, im, arg, Abs, polar_lift, periodic_argument)
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.hyperbolic import cosh, coth, sinh, asinh
from sympy.functions.elementary.miscellaneous import Max, Min, sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import cos, sin, atan
from sympy.functions.special.bessel import besseli, besselj, besselk, bessely
from sympy.functions.special.delta_functions import DiracDelta, Heaviside
from sympy.functions.special.error_functions import erf, erfc, Ei
from sympy.functions.special.gamma_functions import digamma, gamma, lowergamma
from sympy.integrals import integrate, Integral
from sympy.integrals.transforms import (
_simplify, IntegralTransform, IntegralTransformError)
from sympy.logic.boolalg import to_cnf, conjuncts, disjuncts, Or, And
from sympy.matrices.matrices import MatrixBase
from sympy.polys.matrices.linsolve import _lin_eq2dict
from sympy.polys.polyerrors import PolynomialError
from sympy.polys.polyroots import roots
from sympy.polys.polytools import Poly
from sympy.polys.rationaltools import together
from sympy.polys.rootoftools import RootSum
from sympy.utilities.exceptions import (
sympy_deprecation_warning, SymPyDeprecationWarning, ignore_warnings)
from sympy.utilities.misc import debug, debugf
def _simplifyconds(expr, s, a):
r"""
Naively simplify some conditions occurring in ``expr``,
given that `\operatorname{Re}(s) > a`.
Examples
========
>>> from sympy.integrals.laplace import _simplifyconds
>>> from sympy.abc import x
>>> from sympy import sympify as S
>>> _simplifyconds(abs(x**2) < 1, x, 1)
False
>>> _simplifyconds(abs(x**2) < 1, x, 2)
False
>>> _simplifyconds(abs(x**2) < 1, x, 0)
Abs(x**2) < 1
>>> _simplifyconds(abs(1/x**2) < 1, x, 1)
True
>>> _simplifyconds(S(1) < abs(x), x, 1)
True
>>> _simplifyconds(S(1) < abs(1/x), x, 1)
False
>>> from sympy import Ne
>>> _simplifyconds(Ne(1, x**3), x, 1)
True
>>> _simplifyconds(Ne(1, x**3), x, 2)
True
>>> _simplifyconds(Ne(1, x**3), x, 0)
Ne(1, x**3)
"""
def power(ex):
if ex == s:
return 1
if ex.is_Pow and ex.base == s:
return ex.exp
return None
def bigger(ex1, ex2):
""" Return True only if |ex1| > |ex2|, False only if |ex1| < |ex2|.
Else return None. """
if ex1.has(s) and ex2.has(s):
return None
if isinstance(ex1, Abs):
ex1 = ex1.args[0]
if isinstance(ex2, Abs):
ex2 = ex2.args[0]
if ex1.has(s):
return bigger(1/ex2, 1/ex1)
n = power(ex2)
if n is None:
return None
try:
if n > 0 and (Abs(ex1) <= Abs(a)**n) == S.true:
return False
if n < 0 and (Abs(ex1) >= Abs(a)**n) == S.true:
return True
except TypeError:
pass
def replie(x, y):
""" simplify x < y """
if (not (x.is_positive or isinstance(x, Abs))
or not (y.is_positive or isinstance(y, Abs))):
return (x < y)
r = bigger(x, y)
if r is not None:
return not r
return (x < y)
def replue(x, y):
b = bigger(x, y)
if b in (True, False):
return True
return Unequality(x, y)
def repl(ex, *args):
if ex in (True, False):
return bool(ex)
return ex.replace(*args)
from sympy.simplify.radsimp import collect_abs
expr = collect_abs(expr)
expr = repl(expr, Lt, replie)
expr = repl(expr, Gt, lambda x, y: replie(y, x))
expr = repl(expr, Unequality, replue)
return S(expr)
def expand_dirac_delta(expr):
"""
Expand an expression involving DiractDelta to get it as a linear
combination of DiracDelta functions.
"""
return _lin_eq2dict(expr, expr.atoms(DiracDelta))
def _laplace_transform_integration(f, t, s_, simplify=True):
""" The backend function for doing Laplace transforms by integration.
This backend assumes that the frontend has already split sums
such that `f` is to an addition anymore.
"""
s = Dummy('s')
debugf('[LT _l_t_i ] started with (%s, %s, %s)', (f, t, s))
debugf('[LT _l_t_i ] and simplify=%s', (simplify, ))
if f.has(DiracDelta):
return None
F = integrate(f*exp(-s*t), (t, S.Zero, S.Infinity))
debugf('[LT _l_t_i ] integrated: %s', (F, ))
if not F.has(Integral):
return _simplify(F.subs(s, s_), simplify), S.NegativeInfinity, S.true
if not F.is_Piecewise:
debug('[LT _l_t_i ] not piecewise.')
return None
F, cond = F.args[0]
if F.has(Integral):
debug('[LT _l_t_i ] integral in unexpected form.')
return None
def process_conds(conds):
""" Turn ``conds`` into a strip and auxiliary conditions. """
from sympy.solvers.inequalities import _solve_inequality
a = S.NegativeInfinity
aux = S.true
conds = conjuncts(to_cnf(conds))
p, q, w1, w2, w3, w4, w5 = symbols(
'p q w1 w2 w3 w4 w5', cls=Wild, exclude=[s])
patterns = (
p*Abs(arg((s + w3)*q)) < w2,
p*Abs(arg((s + w3)*q)) <= w2,
Abs(periodic_argument((s + w3)**p*q, w1)) < w2,
Abs(periodic_argument((s + w3)**p*q, w1)) <= w2,
Abs(periodic_argument((polar_lift(s + w3))**p*q, w1)) < w2,
Abs(periodic_argument((polar_lift(s + w3))**p*q, w1)) <= w2)
for c in conds:
a_ = S.Infinity
aux_ = []
for d in disjuncts(c):
if d.is_Relational and s in d.rhs.free_symbols:
d = d.reversed
if d.is_Relational and isinstance(d, (Ge, Gt)):
d = d.reversedsign
for pat in patterns:
m = d.match(pat)
if m:
break
if m and m[q].is_positive and m[w2]/m[p] == pi/2:
d = -re(s + m[w3]) < 0
m = d.match(p - cos(w1*Abs(arg(s*w5))*w2)*Abs(s**w3)**w4 < 0)
if not m:
m = d.match(
cos(p - Abs(periodic_argument(s**w1*w5, q))*w2) *
Abs(s**w3)**w4 < 0)
if not m:
m = d.match(
p - cos(
Abs(periodic_argument(polar_lift(s)**w1*w5, q))*w2
)*Abs(s**w3)**w4 < 0)
if m and all(m[wild].is_positive for wild in [
w1, w2, w3, w4, w5]):
d = re(s) > m[p]
d_ = d.replace(
re, lambda x: x.expand().as_real_imag()[0]).subs(re(s), t)
if (
not d.is_Relational or d.rel_op in ('==', '!=')
or d_.has(s) or not d_.has(t)):
aux_ += [d]
continue
soln = _solve_inequality(d_, t)
if not soln.is_Relational or soln.rel_op in ('==', '!='):
aux_ += [d]
continue
if soln.lts == t:
debug('[LT _l_t_i ] convergence not in half-plane.')
return None
else:
a_ = Min(soln.lts, a_)
if a_ is not S.Infinity:
a = Max(a_, a)
else:
aux = And(aux, Or(*aux_))
return a, aux.canonical if aux.is_Relational else aux
conds = [process_conds(c) for c in disjuncts(cond)]
conds2 = [x for x in conds if x[1] !=
S.false and x[0] is not S.NegativeInfinity]
if not conds2:
conds2 = [x for x in conds if x[1] != S.false]
conds = list(ordered(conds2))
def cnt(expr):
if expr in (True, False):
return 0
return expr.count_ops()
conds.sort(key=lambda x: (-x[0], cnt(x[1])))
if not conds:
debug('[LT _l_t_i ] no convergence found.')
return None
a, aux = conds[0] # XXX is [0] always the right one?
def sbs(expr):
return expr.subs(s, s_)
if simplify:
F = _simplifyconds(F, s, a)
aux = _simplifyconds(aux, s, a)
return _simplify(F.subs(s, s_), simplify), sbs(a), _canonical(sbs(aux))
def _laplace_deep_collect(f, t):
"""
This is an internal helper function that traverses through the epression
tree of `f(t)` and collects arguments. The purpose of it is that
anything like `f(w*t-1*t-c)` will be written as `f((w-1)*t-c)` such that
it can match `f(a*t+b)`.
"""
func = f.func
args = list(f.args)
if len(f.args) == 0:
return f
else:
args = [_laplace_deep_collect(arg, t) for arg in args]
if func.is_Add:
return func(*args).collect(t)
else:
return func(*args)
@cacheit
def _laplace_build_rules():
"""
This is an internal helper function that returns the table of Laplace
transform rules in terms of the time variable `t` and the frequency
variable `s`. It is used by ``_laplace_apply_rules``. Each entry is a
tuple containing:
(time domain pattern,
frequency-domain replacement,
condition for the rule to be applied,
convergence plane,
preparation function)
The preparation function is a function with one argument that is applied
to the expression before matching. For most rules it should be
``_laplace_deep_collect``.
"""
t = Dummy('t')
s = Dummy('s')
a = Wild('a', exclude=[t])
b = Wild('b', exclude=[t])
n = Wild('n', exclude=[t])
tau = Wild('tau', exclude=[t])
omega = Wild('omega', exclude=[t])
def dco(f): return _laplace_deep_collect(f, t)
debug('_laplace_build_rules is building rules')
laplace_transform_rules = [
(a, a/s,
S.true, S.Zero, dco), # 4.2.1
(DiracDelta(a*t-b), exp(-s*b/a)/Abs(a),
Or(And(a > 0, b >= 0), And(a < 0, b <= 0)),
S.NegativeInfinity, dco), # Not in Bateman54
(DiracDelta(a*t-b), S(0),
Or(And(a < 0, b >= 0), And(a > 0, b <= 0)),
S.NegativeInfinity, dco), # Not in Bateman54
(Heaviside(a*t-b), exp(-s*b/a)/s,
And(a > 0, b > 0), S.Zero, dco), # 4.4.1
(Heaviside(a*t-b), (1-exp(-s*b/a))/s,
And(a < 0, b < 0), S.Zero, dco), # 4.4.1
(Heaviside(a*t-b), 1/s,
And(a > 0, b <= 0), S.Zero, dco), # 4.4.1
(Heaviside(a*t-b), 0,
And(a < 0, b > 0), S.Zero, dco), # 4.4.1
(t, 1/s**2,
S.true, S.Zero, dco), # 4.2.3
(1/(a*t+b), -exp(-b/a*s)*Ei(-b/a*s)/a,
Abs(arg(b/a)) < pi, S.Zero, dco), # 4.2.6
(1/sqrt(a*t+b), sqrt(a*pi/s)*exp(b/a*s)*erfc(sqrt(b/a*s))/a,
Abs(arg(b/a)) < pi, S.Zero, dco), # 4.2.18
((a*t+b)**(-S(3)/2),
2*b**(-S(1)/2)-2*(pi*s/a)**(S(1)/2)*exp(b/a*s) * erfc(sqrt(b/a*s))/a,
Abs(arg(b/a)) < pi, S.Zero, dco), # 4.2.20
(sqrt(t)/(t+b), sqrt(pi/s)-pi*sqrt(b)*exp(b*s)*erfc(sqrt(b*s)),
Abs(arg(b)) < pi, S.Zero, dco), # 4.2.22
(1/(a*sqrt(t) + t**(3/2)), pi*a**(S(1)/2)*exp(a*s)*erfc(sqrt(a*s)),
S.true, S.Zero, dco), # Not in Bateman54
(t**n, gamma(n+1)/s**(n+1),
n > -1, S.Zero, dco), # 4.3.1
((a*t+b)**n, lowergamma(n+1, b/a*s)*exp(-b/a*s)/s**(n+1)/a,
And(n > -1, Abs(arg(b/a)) < pi), S.Zero, dco), # 4.3.4
(t**n/(t+a), a**n*gamma(n+1)*lowergamma(-n, a*s),
And(n > -1, Abs(arg(a)) < pi), S.Zero, dco), # 4.3.7
(exp(a*t-tau), exp(-tau)/(s-a),
S.true, re(a), dco), # 4.5.1
(t*exp(a*t-tau), exp(-tau)/(s-a)**2,
S.true, re(a), dco), # 4.5.2
(t**n*exp(a*t), gamma(n+1)/(s-a)**(n+1),
re(n) > -1, re(a), dco), # 4.5.3
(exp(-a*t**2), sqrt(pi/4/a)*exp(s**2/4/a)*erfc(s/sqrt(4*a)),
re(a) > 0, S.Zero, dco), # 4.5.21
(t*exp(-a*t**2),
1/(2*a)-2/sqrt(pi)/(4*a)**(S(3)/2)*s*erfc(s/sqrt(4*a)),
re(a) > 0, S.Zero, dco), # 4.5.22
(exp(-a/t), 2*sqrt(a/s)*besselk(1, 2*sqrt(a*s)),
re(a) >= 0, S.Zero, dco), # 4.5.25
(sqrt(t)*exp(-a/t),
S(1)/2*sqrt(pi/s**3)*(1+2*sqrt(a*s))*exp(-2*sqrt(a*s)),
re(a) >= 0, S.Zero, dco), # 4.5.26
(exp(-a/t)/sqrt(t), sqrt(pi/s)*exp(-2*sqrt(a*s)),
re(a) >= 0, S.Zero, dco), # 4.5.27
(exp(-a/t)/(t*sqrt(t)), sqrt(pi/a)*exp(-2*sqrt(a*s)),
re(a) > 0, S.Zero, dco), # 4.5.28
(t**n*exp(-a/t), 2*(a/s)**((n+1)/2)*besselk(n+1, 2*sqrt(a*s)),
re(a) > 0, S.Zero, dco), # 4.5.29
(exp(-2*sqrt(a*t)),
s**(-1)-sqrt(pi*a)*s**(-S(3)/2)*exp(a/s) * erfc(sqrt(a/s)),
Abs(arg(a)) < pi, S.Zero, dco), # 4.5.31
(exp(-2*sqrt(a*t))/sqrt(t), (pi/s)**(S(1)/2)*exp(a/s)*erfc(sqrt(a/s)),
Abs(arg(a)) < pi, S.Zero, dco), # 4.5.33
(log(a*t), -log(exp(S.EulerGamma)*s/a)/s,
a > 0, S.Zero, dco), # 4.6.1
(log(1+a*t), -exp(s/a)/s*Ei(-s/a),
Abs(arg(a)) < pi, S.Zero, dco), # 4.6.4
(log(a*t+b), (log(b)-exp(s/b/a)/s*a*Ei(-s/b))/s*a,
And(a > 0, Abs(arg(b)) < pi), S.Zero, dco), # 4.6.5
(log(t)/sqrt(t), -sqrt(pi/s)*log(4*s*exp(S.EulerGamma)),
S.true, S.Zero, dco), # 4.6.9
(t**n*log(t), gamma(n+1)*s**(-n-1)*(digamma(n+1)-log(s)),
re(n) > -1, S.Zero, dco), # 4.6.11
(log(a*t)**2, (log(exp(S.EulerGamma)*s/a)**2+pi**2/6)/s,
a > 0, S.Zero, dco), # 4.6.13
(sin(omega*t), omega/(s**2+omega**2),
S.true, Abs(im(omega)), dco), # 4,7,1
(Abs(sin(omega*t)), omega/(s**2+omega**2)*coth(pi*s/2/omega),
omega > 0, S.Zero, dco), # 4.7.2
(sin(omega*t)/t, atan(omega/s),
S.true, Abs(im(omega)), dco), # 4.7.16
(sin(omega*t)**2/t, log(1+4*omega**2/s**2)/4,
S.true, 2*Abs(im(omega)), dco), # 4.7.17
(sin(omega*t)**2/t**2,
omega*atan(2*omega/s)-s*log(1+4*omega**2/s**2)/4,
S.true, 2*Abs(im(omega)), dco), # 4.7.20
(sin(2*sqrt(a*t)), sqrt(pi*a)/s/sqrt(s)*exp(-a/s),
S.true, S.Zero, dco), # 4.7.32
(sin(2*sqrt(a*t))/t, pi*erf(sqrt(a/s)),
S.true, S.Zero, dco), # 4.7.34
(cos(omega*t), s/(s**2+omega**2),
S.true, Abs(im(omega)), dco), # 4.7.43
(cos(omega*t)**2, (s**2+2*omega**2)/(s**2+4*omega**2)/s,
S.true, 2*Abs(im(omega)), dco), # 4.7.45
(sqrt(t)*cos(2*sqrt(a*t)), sqrt(pi)/2*s**(-S(5)/2)*(s-2*a)*exp(-a/s),
S.true, S.Zero, dco), # 4.7.66
(cos(2*sqrt(a*t))/sqrt(t), sqrt(pi/s)*exp(-a/s),
S.true, S.Zero, dco), # 4.7.67
(sin(a*t)*sin(b*t), 2*a*b*s/(s**2+(a+b)**2)/(s**2+(a-b)**2),
S.true, Abs(im(a))+Abs(im(b)), dco), # 4.7.78
(cos(a*t)*sin(b*t), b*(s**2-a**2+b**2)/(s**2+(a+b)**2)/(s**2+(a-b)**2),
S.true, Abs(im(a))+Abs(im(b)), dco), # 4.7.79
(cos(a*t)*cos(b*t), s*(s**2+a**2+b**2)/(s**2+(a+b)**2)/(s**2+(a-b)**2),
S.true, Abs(im(a))+Abs(im(b)), dco), # 4.7.80
(sinh(a*t), a/(s**2-a**2),
S.true, Abs(re(a)), dco), # 4.9.1
(cosh(a*t), s/(s**2-a**2),
S.true, Abs(re(a)), dco), # 4.9.2
(sinh(a*t)**2, 2*a**2/(s**3-4*a**2*s),
S.true, 2*Abs(re(a)), dco), # 4.9.3
(cosh(a*t)**2, (s**2-2*a**2)/(s**3-4*a**2*s),
S.true, 2*Abs(re(a)), dco), # 4.9.4
(sinh(a*t)/t, log((s+a)/(s-a))/2,
S.true, Abs(re(a)), dco), # 4.9.12
(t**n*sinh(a*t), gamma(n+1)/2*((s-a)**(-n-1)-(s+a)**(-n-1)),
n > -2, Abs(a), dco), # 4.9.18
(t**n*cosh(a*t), gamma(n+1)/2*((s-a)**(-n-1)+(s+a)**(-n-1)),
n > -1, Abs(a), dco), # 4.9.19
(sinh(2*sqrt(a*t)), sqrt(pi*a)/s/sqrt(s)*exp(a/s),
S.true, S.Zero, dco), # 4.9.34
(cosh(2*sqrt(a*t)), 1/s+sqrt(pi*a)/s/sqrt(s)*exp(a/s)*erf(sqrt(a/s)),
S.true, S.Zero, dco), # 4.9.35
(
sqrt(t)*sinh(2*sqrt(a*t)),
pi**(S(1)/2)*s**(-S(5)/2)*(s/2+a) *
exp(a/s)*erf(sqrt(a/s))-a**(S(1)/2)*s**(-2),
S.true, S.Zero, dco), # 4.9.36
(sqrt(t)*cosh(2*sqrt(a*t)), pi**(S(1)/2)*s**(-S(5)/2)*(s/2+a)*exp(a/s),
S.true, S.Zero, dco), # 4.9.37
(sinh(2*sqrt(a*t))/sqrt(t),
pi**(S(1)/2)*s**(-S(1)/2)*exp(a/s) * erf(sqrt(a/s)),
S.true, S.Zero, dco), # 4.9.38
(cosh(2*sqrt(a*t))/sqrt(t), pi**(S(1)/2)*s**(-S(1)/2)*exp(a/s),
S.true, S.Zero, dco), # 4.9.39
(sinh(sqrt(a*t))**2/sqrt(t), pi**(S(1)/2)/2*s**(-S(1)/2)*(exp(a/s)-1),
S.true, S.Zero, dco), # 4.9.40
(cosh(sqrt(a*t))**2/sqrt(t), pi**(S(1)/2)/2*s**(-S(1)/2)*(exp(a/s)+1),
S.true, S.Zero, dco), # 4.9.41
(erf(a*t), exp(s**2/(2*a)**2)*erfc(s/(2*a))/s,
4*Abs(arg(a)) < pi, S.Zero, dco), # 4.12.2
(erf(sqrt(a*t)), sqrt(a)/sqrt(s+a)/s,
S.true, Max(S.Zero, -re(a)), dco), # 4.12.4
(exp(a*t)*erf(sqrt(a*t)), sqrt(a)/sqrt(s)/(s-a),
S.true, Max(S.Zero, re(a)), dco), # 4.12.5
(erf(sqrt(a/t)/2), (1-exp(-sqrt(a*s)))/s,
re(a) > 0, S.Zero, dco), # 4.12.6
(erfc(sqrt(a*t)), (sqrt(s+a)-sqrt(a))/sqrt(s+a)/s,
S.true, -re(a), dco), # 4.12.9
(exp(a*t)*erfc(sqrt(a*t)), 1/(s+sqrt(a*s)),
S.true, S.Zero, dco), # 4.12.10
(erfc(sqrt(a/t)/2), exp(-sqrt(a*s))/s,
re(a) > 0, S.Zero, dco), # 4.2.11
(besselj(n, a*t), a**n/(sqrt(s**2+a**2)*(s+sqrt(s**2+a**2))**n),
re(n) > -1, Abs(im(a)), dco), # 4.14.1
(t**b*besselj(n, a*t),
2**n/sqrt(pi)*gamma(n+S.Half)*a**n*(s**2+a**2)**(-n-S.Half),
And(re(n) > -S.Half, Eq(b, n)), Abs(im(a)), dco), # 4.14.7
(t**b*besselj(n, a*t),
2**(n+1)/sqrt(pi)*gamma(n+S(3)/2)*a**n*s*(s**2+a**2)**(-n-S(3)/2),
And(re(n) > -1, Eq(b, n+1)), Abs(im(a)), dco), # 4.14.8
(besselj(0, 2*sqrt(a*t)), exp(-a/s)/s,
S.true, S.Zero, dco), # 4.14.25
(t**(b)*besselj(n, 2*sqrt(a*t)), a**(n/2)*s**(-n-1)*exp(-a/s),
And(re(n) > -1, Eq(b, n*S.Half)), S.Zero, dco), # 4.14.30
(besselj(0, a*sqrt(t**2+b*t)),
exp(b*s-b*sqrt(s**2+a**2))/sqrt(s**2+a**2),
Abs(arg(b)) < pi, Abs(im(a)), dco), # 4.15.19
(besseli(n, a*t), a**n/(sqrt(s**2-a**2)*(s+sqrt(s**2-a**2))**n),
re(n) > -1, Abs(re(a)), dco), # 4.16.1
(t**b*besseli(n, a*t),
2**n/sqrt(pi)*gamma(n+S.Half)*a**n*(s**2-a**2)**(-n-S.Half),
And(re(n) > -S.Half, Eq(b, n)), Abs(re(a)), dco), # 4.16.6
(t**b*besseli(n, a*t),
2**(n+1)/sqrt(pi)*gamma(n+S(3)/2)*a**n*s*(s**2-a**2)**(-n-S(3)/2),
And(re(n) > -1, Eq(b, n+1)), Abs(re(a)), dco), # 4.16.7
(t**(b)*besseli(n, 2*sqrt(a*t)), a**(n/2)*s**(-n-1)*exp(a/s),
And(re(n) > -1, Eq(b, n*S.Half)), S.Zero, dco), # 4.16.18
(bessely(0, a*t), -2/pi*asinh(s/a)/sqrt(s**2+a**2),
S.true, Abs(im(a)), dco), # 4.15.44
(besselk(0, a*t), log((s + sqrt(s**2-a**2))/a)/(sqrt(s**2-a**2)),
S.true, -re(a), dco) # 4.16.23
]
return laplace_transform_rules, t, s
def _laplace_rule_timescale(f, t, s):
"""
This function applies the time-scaling rule of the Laplace transform in
a straight-forward way. For example, if it gets ``(f(a*t), t, s)``, it will
compute ``LaplaceTransform(f(t)/a, t, s/a)`` if ``a>0``.
"""
a = Wild('a', exclude=[t])
g = WildFunction('g', nargs=1)
ma1 = f.match(g)
if ma1:
arg = ma1[g].args[0].collect(t)
ma2 = arg.match(a*t)
if ma2 and ma2[a].is_positive and ma2[a] != 1:
debug('_laplace_apply_prog rules match:')
debugf(' f: %s _ %s, %s )', (f, ma1, ma2))
debug(' rule: time scaling (4.1.4)')
r, pr, cr = _laplace_transform(1/ma2[a]*ma1[g].func(t),
t, s/ma2[a], simplify=False)
return (r, pr, cr)
return None
def _laplace_rule_heaviside(f, t, s):
"""
This function deals with time-shifted Heaviside step functions. If the time
shift is positive, it applies the time-shift rule of the Laplace transform.
For example, if it gets ``(Heaviside(t-a)*f(t), t, s)``, it will compute
``exp(-a*s)*LaplaceTransform(f(t+a), t, s)``.
If the time shift is negative, the Heaviside function is simply removed
as it means nothing to the Laplace transform.
The function does not remove a factor ``Heaviside(t)``; this is done by
the simple rules.
"""
a = Wild('a', exclude=[t])
y = Wild('y')
g = Wild('g')
ma1 = f.match(Heaviside(y)*g)
if ma1:
ma2 = ma1[y].match(t-a)
if ma2 and ma2[a].is_positive:
debug('_laplace_apply_prog_rules match:')
debugf(' f: %s ( %s, %s )', (f, ma1, ma2))
debug(' rule: time shift (4.1.4)')
r, pr, cr = _laplace_transform(ma1[g].subs(t, t+ma2[a]), t, s,
simplify=False)
return (exp(-ma2[a]*s)*r, pr, cr)
if ma2 and ma2[a].is_negative:
debug('_laplace_apply_prog_rules match:')
debugf(' f: %s ( %s, %s )', (f, ma1, ma2))
debug(' rule: Heaviside factor, negative time shift (4.1.4)')
r, pr, cr = _laplace_transform(ma1[g], t, s, simplify=False)
return (r, pr, cr)
return None
def _laplace_rule_exp(f, t, s):
"""
If this function finds a factor ``exp(a*t)``, it applies the
frequency-shift rule of the Laplace transform and adjusts the convergence
plane accordingly. For example, if it gets ``(exp(-a*t)*f(t), t, s)``, it
will compute ``LaplaceTransform(f(t), t, s+a)``.
"""
a = Wild('a', exclude=[t])
y = Wild('y')
z = Wild('z')
ma1 = f.match(exp(y)*z)
if ma1:
ma2 = ma1[y].collect(t).match(a*t)
if ma2:
debug('_laplace_apply_prog_rules match:')
debugf(' f: %s ( %s, %s )', (f, ma1, ma2))
debug(' rule: multiply with exp (4.1.5)')
r, pr, cr = _laplace_transform(ma1[z], t, s-ma2[a],
simplify=False)
return (r, pr+re(ma2[a]), cr)
return None
def _laplace_rule_delta(f, t, s):
"""
If this function finds a factor ``DiracDelta(b*t-a)``, it applies the
masking property of the delta distribution. For example, if it gets
``(DiracDelta(t-a)*f(t), t, s)``, it will return
``(f(a)*exp(-a*s), -a, True)``.
"""
# This rule is not in Bateman54
a = Wild('a', exclude=[t])
b = Wild('b', exclude=[t])
y = Wild('y')
z = Wild('z')
ma1 = f.match(DiracDelta(y)*z)
if ma1 and not ma1[z].has(DiracDelta):
ma2 = ma1[y].collect(t).match(b*t-a)
if ma2:
debug('_laplace_apply_prog_rules match:')
debugf(' f: %s ( %s, %s )', (f, ma1, ma2))
debug(' rule: multiply with DiracDelta')
loc = ma2[a]/ma2[b]
if re(loc) >= 0 and im(loc) == 0:
r = exp(-ma2[a]/ma2[b]*s)*ma1[z].subs(t, ma2[a]/ma2[b])/ma2[b]
return (r, S.NegativeInfinity, S.true)
else:
return (0, S.NegativeInfinity, S.true)
if ma1[y].is_polynomial(t):
ro = roots(ma1[y], t)
if roots is not {} and set(ro.values()) == {1}:
slope = diff(ma1[y], t)
r = Add(
*[exp(-x*s)*ma1[z].subs(t, s)/slope.subs(t, x)
for x in list(ro.keys()) if im(x) == 0 and re(x) >= 0])
return (r, S.NegativeInfinity, S.true)
return None
def _laplace_trig_split(fn):
"""
Helper function for `_laplace_rule_trig`. This function returns two terms
`f` and `g`. `f` contains all product terms with sin, cos, sinh, cosh in
them; `g` contains everything else.
"""
trigs = [S.One]
other = [S.One]
for term in Mul.make_args(fn):
if term.has(sin, cos, sinh, cosh, exp):
trigs.append(term)
else:
other.append(term)
f = Mul(*trigs)
g = Mul(*other)
return f, g
def _laplace_trig_expsum(f, t):
"""
Helper function for `_laplace_rule_trig`. This function expects the `f`
from `_laplace_trig_split`. It returns two lists `xm` and `xn`. `xm` is
a list of dictionaries with keys `k` and `a` representing a function
`k*exp(a*t)`. `xn` is a list of all terms that cannot be brought into
that form, which may happen, e.g., when a trigonometric function has
another function in its argument.
"""
m = Wild('m')
p = Wild('p', exclude=[t])
xm = []
xn = []
x1 = f.rewrite(exp).expand()
for term in Add.make_args(x1):
if not term.has(t):
xm.append({'k': term, 'a': 0, re: 0, im: 0})
continue
term = term.powsimp(combine='exp')
if (r := term.match(p*exp(m))) is not None:
if (mp := r[m].as_poly(t)) is not None:
mc = mp.all_coeffs()
if len(mc) == 2:
xm.append({
'k': r[p]*exp(mc[1]), 'a': mc[0],
re: re(mc[0]), im: im(mc[0])})
else:
xn.append(term)
else:
xn.append(term)
else:
xn.append(term)
return xm, xn
def _laplace_trig_ltex(xm, t, s):
"""
Helper function for `_laplace_rule_trig`. This function takes the list of
exponentials `xm` from `_laplace_trig_expsum` and simplifies complex
conjugate and real symmetric poles. It returns the result as a sum and
the convergence plane.
"""
results = []
planes = []
def _simpc(coeffs):
nc = coeffs.copy()
for k in range(len(nc)):
ri = nc[k].as_real_imag()
if ri[0].has(im):
nc[k] = nc[k].rewrite(cos)
else:
nc[k] = (ri[0] + I*ri[1]).rewrite(cos)
return nc
def _quadpole(t1, k1, k2, k3, s):
a, k0, a_r, a_i = t1['a'], t1['k'], t1[re], t1[im]
nc = [
k0 + k1 + k2 + k3,
a*(k0 + k1 - k2 - k3) - 2*I*a_i*k1 + 2*I*a_i*k2,
(
a**2*(-k0 - k1 - k2 - k3) +
a*(4*I*a_i*k0 + 4*I*a_i*k3) +
4*a_i**2*k0 + 4*a_i**2*k3),
(
a**3*(-k0 - k1 + k2 + k3) +
a**2*(4*I*a_i*k0 + 2*I*a_i*k1 - 2*I*a_i*k2 - 4*I*a_i*k3) +
a*(4*a_i**2*k0 - 4*a_i**2*k3))
]
dc = [
S.One, S.Zero, 2*a_i**2 - 2*a_r**2,
S.Zero, a_i**4 + 2*a_i**2*a_r**2 + a_r**4]
n = Add(
*[x*s**y for x, y in zip(_simpc(nc), range(len(nc))[::-1])])
d = Add(
*[x*s**y for x, y in zip(dc, range(len(dc))[::-1])])
debugf(' quadpole: (%s) / (%s)', (n, d))
return n/d
def _ccpole(t1, k1, s):
a, k0, a_r, a_i = t1['a'], t1['k'], t1[re], t1[im]
nc = [k0 + k1, -a*k0 - a*k1 + 2*I*a_i*k0]
dc = [S.One, -2*a_r, a_i**2 + a_r**2]
n = Add(
*[x*s**y for x, y in zip(_simpc(nc), range(len(nc))[::-1])])
d = Add(
*[x*s**y for x, y in zip(dc, range(len(dc))[::-1])])
debugf(' ccpole: (%s) / (%s)', (n, d))
return n/d
def _rspole(t1, k2, s):
a, k0, a_r, a_i = t1['a'], t1['k'], t1[re], t1[im]
nc = [k0 + k2, a*k0 - a*k2 - 2*I*a_i*k0]
dc = [S.One, -2*I*a_i, -a_i**2 - a_r**2]
n = Add(
*[x*s**y for x, y in zip(_simpc(nc), range(len(nc))[::-1])])
d = Add(
*[x*s**y for x, y in zip(dc, range(len(dc))[::-1])])
debugf(' rspole: (%s) / (%s)', (n, d))
return n/d
def _sypole(t1, k3, s):
a, k0 = t1['a'], t1['k']
nc = [k0 + k3, a*(k0 - k3)]
dc = [S.One, S.Zero, -a**2]
n = Add(
*[x*s**y for x, y in zip(_simpc(nc), range(len(nc))[::-1])])
d = Add(
*[x*s**y for x, y in zip(dc, range(len(dc))[::-1])])
debugf(' sypole: (%s) / (%s)', (n, d))
return n/d
def _simplepole(t1, s):
a, k0 = t1['a'], t1['k']
n = k0
d = s - a
debugf(' simplepole: (%s) / (%s)', (n, d))
return n/d
while len(xm) > 0:
t1 = xm.pop()
i_imagsym = None
i_realsym = None
i_pointsym = None
# The following code checks all remaining poles. If t1 is a pole at
# a+b*I, then we check for a-b*I, -a+b*I, and -a-b*I, and
# assign the respective indices to i_imagsym, i_realsym, i_pointsym.
# -a-b*I / i_pointsym only applies if both a and b are != 0.
for i in range(len(xm)):
real_eq = t1[re] == xm[i][re]
realsym = t1[re] == -xm[i][re]
imag_eq = t1[im] == xm[i][im]
imagsym = t1[im] == -xm[i][im]
if realsym and imagsym and t1[re] != 0 and t1[im] != 0:
i_pointsym = i
elif realsym and imag_eq and t1[re] != 0:
i_realsym = i
elif real_eq and imagsym and t1[im] != 0:
i_imagsym = i
# The next part looks for four possible pole constellations:
# quad: a+b*I, a-b*I, -a+b*I, -a-b*I
# cc: a+b*I, a-b*I (a may be zero)
# quad: a+b*I, -a+b*I (b may be zero)
# point: a+b*I, -a-b*I (a!=0 and b!=0 is needed, but that has been
# asserted when finding i_pointsym above.)
# If none apply, then t1 is a simple pole.
if (
i_imagsym is not None and i_realsym is not None
and i_pointsym is not None):
results.append(
_quadpole(t1,
xm[i_imagsym]['k'], xm[i_realsym]['k'],
xm[i_pointsym]['k'], s))
planes.append(Abs(re(t1['a'])))
# The three additional poles have now been used; to pop them
# easily we have to do it from the back.
indices_to_pop = [i_imagsym, i_realsym, i_pointsym]
indices_to_pop.sort(reverse=True)
for i in indices_to_pop:
xm.pop(i)
elif i_imagsym is not None:
results.append(_ccpole(t1, xm[i_imagsym]['k'], s))
planes.append(t1[re])
xm.pop(i_imagsym)
elif i_realsym is not None:
results.append(_rspole(t1, xm[i_realsym]['k'], s))
planes.append(Abs(t1[re]))
xm.pop(i_realsym)
elif i_pointsym is not None:
results.append(_sypole(t1, xm[i_pointsym]['k'], s))
planes.append(Abs(t1[re]))
xm.pop(i_pointsym)
else:
results.append(_simplepole(t1, s))
planes.append(t1[re])
return Add(*results), Max(*planes)
def _laplace_rule_trig(fn, t_, s, doit=True, **hints):
"""
This rule covers trigonometric factors by splitting everything into a
sum of exponential functions and collecting complex conjugate poles and
real symmetric poles.
"""
t = Dummy('t', real=True)
if not fn.has(sin, cos, sinh, cosh):
return None
debugf('_laplace_rule_trig: (%s, %s, %s)', (fn, t_, s))
f, g = _laplace_trig_split(fn.subs(t_, t))
debugf(' f = %s\n g = %s', (f, g))
xm, xn = _laplace_trig_expsum(f, t)
debugf(' xm = %s\n xn = %s', (xm, xn))
if len(xn) > 0:
# not implemented yet
debug(' --> xn is not empty; giving up.')
return None
if not g.has(t):
r, p = _laplace_trig_ltex(xm, t, s)
return g*r, p, S.true
else:
# Just transform `g` and make frequency-shifted copies
planes = []
results = []
G, G_plane, G_cond = _laplace_transform(g, t, s)
for x1 in xm:
results.append(x1['k']*G.subs(s, s-x1['a']))
planes.append(G_plane+re(x1['a']))
return Add(*results).subs(t, t_), Max(*planes), G_cond
def _laplace_rule_diff(f, t, s, doit=True, **hints):
"""
This function looks for derivatives in the time domain and replaces it
by factors of `s` and initial conditions in the frequency domain. For
example, if it gets ``(diff(f(t), t), t, s)``, it will compute
``s*LaplaceTransform(f(t), t, s) - f(0)``.
"""
a = Wild('a', exclude=[t])
n = Wild('n', exclude=[t])
g = WildFunction('g')
ma1 = f.match(a*Derivative(g, (t, n)))
if ma1 and ma1[n].is_integer:
m = [z.has(t) for z in ma1[g].args]
if sum(m) == 1:
debug('_laplace_apply_rules match:')
debugf(' f, n: %s, %s', (f, ma1[n]))
debug(' rule: time derivative (4.1.8)')
d = []
for k in range(ma1[n]):
if k == 0:
y = ma1[g].subs(t, 0)
else:
y = Derivative(ma1[g], (t, k)).subs(t, 0)
d.append(s**(ma1[n]-k-1)*y)
r, pr, cr = _laplace_transform(ma1[g], t, s, simplify=False)
return (ma1[a]*(s**ma1[n]*r - Add(*d)), pr, cr)
return None
def _laplace_rule_sdiff(f, t, s, doit=True, **hints):
"""
This function looks for multiplications with polynoimials in `t` as they
correspond to differentiation in the frequency domain. For example, if it
gets ``(t*f(t), t, s)``, it will compute
``-Derivative(LaplaceTransform(f(t), t, s), s)``.
"""
if f.is_Mul:
pfac = [1]
ofac = [1]
for fac in Mul.make_args(f):
if fac.is_polynomial(t):
pfac.append(fac)
else:
ofac.append(fac)
if len(pfac) > 1:
pex = prod(pfac)
pc = Poly(pex, t).all_coeffs()
N = len(pc)
if N > 1:
debug('_laplace_apply_rules match:')
debugf(' f, n: %s, %s', (f, pfac))
debug(' rule: frequency derivative (4.1.6)')
oex = prod(ofac)
r_, p_, c_ = _laplace_transform(oex, t, s, simplify=False)
deri = [r_]
d1 = False
try:
d1 = -diff(deri[-1], s)
except ValueError:
d1 = False
if r_.has(LaplaceTransform):
for k in range(N-1):
deri.append((-1)**(k+1)*Derivative(r_, s, k+1))
else:
if d1:
deri.append(d1)
for k in range(N-2):
deri.append(-diff(deri[-1], s))
if d1:
r = Add(*[pc[N-n-1]*deri[n] for n in range(N)])
return (r, p_, c_)
return None
def _laplace_expand(f, t, s, doit=True, **hints):
"""
This function tries to expand its argument with successively stronger
methods: first it will expand on the top level, then it will expand any
multiplications in depth, then it will try all avilable expansion methods,
and finally it will try to expand trigonometric functions.
If it can expand, it will then compute the Laplace transform of the
expanded term.
"""
if f.is_Add:
return None
r = expand(f, deep=False)
if r.is_Add:
return _laplace_transform(r, t, s, simplify=False)
r = expand_mul(f)
if r.is_Add:
return _laplace_transform(r, t, s, simplify=False)
r = expand(f)
if r.is_Add:
return _laplace_transform(r, t, s, simplify=False)
if r != f:
return _laplace_transform(r, t, s, simplify=False)
r = expand(expand_trig(f))
if r.is_Add:
return _laplace_transform(r, t, s, simplify=False)
return None
def _laplace_apply_prog_rules(f, t, s):
"""
This function applies all program rules and returns the result if one
of them gives a result.
"""
prog_rules = [_laplace_rule_heaviside, _laplace_rule_delta,
_laplace_rule_timescale, _laplace_rule_exp,
_laplace_rule_trig,
_laplace_rule_diff, _laplace_rule_sdiff]
for p_rule in prog_rules:
if (L := p_rule(f, t, s)) is not None:
return L
return None
def _laplace_apply_simple_rules(f, t, s):
"""
This function applies all simple rules and returns the result if one
of them gives a result.
"""
simple_rules, t_, s_ = _laplace_build_rules()
prep_old = ''
prep_f = ''
for t_dom, s_dom, check, plane, prep in simple_rules:
if prep_old != prep:
prep_f = prep(f.subs({t: t_}))
prep_old = prep
ma = prep_f.match(t_dom)
if ma:
try:
c = check.xreplace(ma)
except TypeError:
# This may happen if the time function has imaginary
# numbers in it. Then we give up.
continue
if c == S.true:
debug('_laplace_apply_simple_rules match:')
debugf(' f: %s', (f,))
debugf(' rule: %s o---o %s', (t_dom, s_dom))
debugf(' match: %s', (ma, ))
return (s_dom.xreplace(ma).subs({s_: s}),
plane.xreplace(ma), S.true)
return None
def _laplace_transform(fn, t_, s_, simplify=True):
"""
Front-end function of the Laplace transform. It tries to apply all known
rules recursively, and if everything else fails, it tries to integrate.
"""
debugf('[LT _l_t] (%s, %s, %s)', (fn, t_, s_))
terms = Add.make_args(fn)
terms_s = []
planes = []
conditions = []
for ff in terms:
k, ft = ff.as_independent(t_, as_Add=False)
if (r := _laplace_apply_simple_rules(ft, t_, s_)) is not None:
pass
elif (r := _laplace_apply_prog_rules(ft, t_, s_)) is not None:
pass
elif (r := _laplace_expand(ft, t_, s_)) is not None:
pass
elif any(undef.has(t_) for undef in ft.atoms(AppliedUndef)):
# If there are undefined functions f(t) then integration is
# unlikely to do anything useful so we skip it and given an
# unevaluated LaplaceTransform.
r = (LaplaceTransform(ft, t_, s_), S.NegativeInfinity, True)
elif (r := _laplace_transform_integration(
ft, t_, s_, simplify=simplify)) is not None:
pass
else:
r = (LaplaceTransform(ft, t_, s_), S.NegativeInfinity, True)
(ri_, pi_, ci_) = r
terms_s.append(k*ri_)
planes.append(pi_)
conditions.append(ci_)
result = Add(*terms_s)
if simplify:
result = result.simplify(doit=False)
plane = Max(*planes)
condition = And(*conditions)
return result, plane, condition
class LaplaceTransform(IntegralTransform):
"""
Class representing unevaluated Laplace transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute Laplace transforms, see the :func:`laplace_transform`
docstring.
If this is called with ``.doit()``, it returns the Laplace transform as an
expression. If it is called with ``.doit(noconds=False)``, it returns a
tuple containing the same expression, a convergence plane, and conditions.
"""
_name = 'Laplace'
def _compute_transform(self, f, t, s, **hints):
_simplify = hints.get('simplify', False)
LT = _laplace_transform_integration(f, t, s, simplify=_simplify)
return LT
def _as_integral(self, f, t, s):
return Integral(f*exp(-s*t), (t, S.Zero, S.Infinity))
def _collapse_extra(self, extra):
conds = []
planes = []
for plane, cond in extra:
conds.append(cond)
planes.append(plane)
cond = And(*conds)
plane = Max(*planes)
if cond == S.false:
raise IntegralTransformError(
'Laplace', None, 'No combined convergence.')
return plane, cond
def doit(self, **hints):
"""
Try to evaluate the transform in closed form.
Explanation
===========
Standard hints are the following:
- ``noconds``: if True, do not return convergence conditions. The
default setting is `True`.
- ``simplify``: if True, it simplifies the final result. The
default setting is `False`.
"""
_noconds = hints.get('noconds', True)
_simplify = hints.get('simplify', False)
debugf('[LT doit] (%s, %s, %s)', (self.function,
self.function_variable,
self.transform_variable))
t_ = self.function_variable
s_ = self.transform_variable
fn = self.function
r = _laplace_transform(fn, t_, s_, simplify=_simplify)
if _noconds:
return r[0]
else:
return r
def laplace_transform(f, t, s, legacy_matrix=True, **hints):
r"""
Compute the Laplace Transform `F(s)` of `f(t)`,
.. math :: F(s) = \int_{0^{-}}^\infty e^{-st} f(t) \mathrm{d}t.
Explanation
===========
For all sensible functions, this converges absolutely in a
half-plane
.. math :: a < \operatorname{Re}(s)
This function returns ``(F, a, cond)`` where ``F`` is the Laplace
transform of ``f``, `a` is the half-plane of convergence, and `cond` are
auxiliary convergence conditions.
The implementation is rule-based, and if you are interested in which
rules are applied, and whether integration is attempted, you can switch
debug information on by setting ``sympy.SYMPY_DEBUG=True``. The numbers
of the rules in the debug information (and the code) refer to Bateman's
Tables of Integral Transforms [1].
The lower bound is `0-`, meaning that this bound should be approached
from the lower side. This is only necessary if distributions are involved.
At present, it is only done if `f(t)` contains ``DiracDelta``, in which
case the Laplace transform is computed implicitly as
.. math ::
F(s) = \lim_{\tau\to 0^{-}} \int_{\tau}^\infty e^{-st}
f(t) \mathrm{d}t
by applying rules.
If the Laplace transform cannot be fully computed in closed form, this
function returns expressions containing unevaluated
:class:`LaplaceTransform` objects.
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`. If
``noconds=True``, only `F` will be returned (i.e. not ``cond``, and also
not the plane ``a``).
.. deprecated:: 1.9
Legacy behavior for matrices where ``laplace_transform`` with
``noconds=False`` (the default) returns a Matrix whose elements are
tuples. The behavior of ``laplace_transform`` for matrices will change
in a future release of SymPy to return a tuple of the transformed
Matrix and the convergence conditions for the matrix as a whole. Use
``legacy_matrix=False`` to enable the new behavior.
Examples
========
>>> from sympy import DiracDelta, exp, laplace_transform
>>> from sympy.abc import t, s, a
>>> laplace_transform(t**4, t, s)
(24/s**5, 0, True)
>>> laplace_transform(t**a, t, s)
(gamma(a + 1)/(s*s**a), 0, re(a) > -1)
>>> laplace_transform(DiracDelta(t)-a*exp(-a*t), t, s, simplify=True)
(s/(a + s), -re(a), True)
References
==========
.. [1] Erdelyi, A. (ed.), Tables of Integral Transforms, Volume 1,
Bateman Manuscript Prooject, McGraw-Hill (1954), available:
https://resolver.caltech.edu/CaltechAUTHORS:20140123-101456353
See Also
========
inverse_laplace_transform, mellin_transform, fourier_transform
hankel_transform, inverse_hankel_transform
"""
_noconds = hints.get('noconds', False)
_simplify = hints.get('simplify', False)
if isinstance(f, MatrixBase) and hasattr(f, 'applyfunc'):
conds = not hints.get('noconds', False)
if conds and legacy_matrix:
adt = 'deprecated-laplace-transform-matrix'
sympy_deprecation_warning(
"""
Calling laplace_transform() on a Matrix with noconds=False (the default) is
deprecated. Either noconds=True or use legacy_matrix=False to get the new
behavior.
""",
deprecated_since_version='1.9',
active_deprecations_target=adt,
)
# Temporarily disable the deprecation warning for non-Expr objects
# in Matrix
with ignore_warnings(SymPyDeprecationWarning):
return f.applyfunc(
lambda fij: laplace_transform(fij, t, s, **hints))
else:
elements_trans = [laplace_transform(
fij, t, s, **hints) for fij in f]
if conds:
elements, avals, conditions = zip(*elements_trans)
f_laplace = type(f)(*f.shape, elements)
return f_laplace, Max(*avals), And(*conditions)
else:
return type(f)(*f.shape, elements_trans)
LT = LaplaceTransform(f, t, s).doit(noconds=False, simplify=_simplify)
if not _noconds:
return LT
else:
return LT[0]
def _inverse_laplace_transform_integration(F, s, t_, plane, simplify=True):
""" The backend function for inverse Laplace transforms. """
from sympy.integrals.meijerint import meijerint_inversion, _get_coeff_exp
from sympy.integrals.transforms import inverse_mellin_transform
# There are two strategies we can try:
# 1) Use inverse mellin transform, related by a simple change of variables.
# 2) Use the inversion integral.
t = Dummy('t', real=True)
def pw_simp(*args):
""" Simplify a piecewise expression from hyperexpand. """
# XXX we break modularity here!
if len(args) != 3:
return Piecewise(*args)
arg = args[2].args[0].argument
coeff, exponent = _get_coeff_exp(arg, t)
e1 = args[0].args[0]
e2 = args[1].args[0]
return (
Heaviside(1/Abs(coeff) - t**exponent)*e1 +
Heaviside(t**exponent - 1/Abs(coeff))*e2)
if F.is_rational_function(s):
F = F.apart(s)
if F.is_Add:
f = Add(
*[_inverse_laplace_transform_integration(X, s, t, plane, simplify)
for X in F.args])
return _simplify(f.subs(t, t_), simplify), True
try:
f, cond = inverse_mellin_transform(F, s, exp(-t), (None, S.Infinity),
needeval=True, noconds=False)
except IntegralTransformError:
f = None
if f is None:
f = meijerint_inversion(F, s, t)
if f is None:
return None
if f.is_Piecewise:
f, cond = f.args[0]
if f.has(Integral):
return None
else:
cond = S.true
f = f.replace(Piecewise, pw_simp)
if f.is_Piecewise:
# many of the functions called below can't work with piecewise
# (b/c it has a bool in args)
return f.subs(t, t_), cond
u = Dummy('u')
def simp_heaviside(arg, H0=S.Half):
a = arg.subs(exp(-t), u)
if a.has(t):
return Heaviside(arg, H0)
from sympy.solvers.inequalities import _solve_inequality
rel = _solve_inequality(a > 0, u)
if rel.lts == u:
k = log(rel.gts)
return Heaviside(t + k, H0)
else:
k = log(rel.lts)
return Heaviside(-(t + k), H0)
f = f.replace(Heaviside, simp_heaviside)
def simp_exp(arg):
return expand_complex(exp(arg))
f = f.replace(exp, simp_exp)
# TODO it would be nice to fix cosh and sinh ... simplify messes these
# exponentials up
return _simplify(f.subs(t, t_), simplify), cond
def _complete_the_square_in_denom(f, s):
from sympy.simplify.radsimp import fraction
[n, d] = fraction(f)
if d.is_polynomial(s):
cf = d.as_poly(s).all_coeffs()
if len(cf) == 3:
a, b, c = cf
d = a*((s+b/(2*a))**2+c/a-(b/(2*a))**2)
return n/d
@cacheit
def _inverse_laplace_build_rules():
"""
This is an internal helper function that returns the table of inverse
Laplace transform rules in terms of the time variable `t` and the
frequency variable `s`. It is used by `_inverse_laplace_apply_rules`.
"""
s = Dummy('s')
t = Dummy('t')
a = Wild('a', exclude=[s])
b = Wild('b', exclude=[s])
c = Wild('c', exclude=[s])
debug('_inverse_laplace_build_rules is building rules')
def _frac(f, s):
try:
return f.factor(s)
except PolynomialError:
return f
def same(f): return f
# This list is sorted according to the prep function needed.
_ILT_rules = [
(a/s, a, S.true, same, 1),
(b*(s+a)**(-c), t**(c-1)*exp(-a*t)/gamma(c), c > 0, same, 1),
(1/(s**2+a**2)**2, (sin(a*t) - a*t*cos(a*t))/(2*a**3),
S.true, same, 1),
# The next two rules must be there in that order. For the second
# one, the condition would be a != 0 or, respectively, to take the
# limit a -> 0 after the transform if a == 0. It is much simpler if
# the case a == 0 has its own rule.
(1/(s**b), t**(b - 1)/gamma(b), S.true, same, 1),
(1/(s*(s+a)**b), lowergamma(b, a*t)/(a**b*gamma(b)),
S.true, same, 1)
]
return _ILT_rules, s, t
def _inverse_laplace_apply_simple_rules(f, s, t):
"""
Helper function for the class InverseLaplaceTransform.
"""
if f == 1:
debug('_inverse_laplace_apply_simple_rules match:')
debugf(' f: %s', (1,))
debugf(' rule: 1 o---o DiracDelta(%s)', (t,))
return DiracDelta(t), S.true
_ILT_rules, s_, t_ = _inverse_laplace_build_rules()
_prep = ''
fsubs = f.subs({s: s_})
for s_dom, t_dom, check, prep, fac in _ILT_rules:
if _prep != (prep, fac):
_F = prep(fsubs*fac)
_prep = (prep, fac)
ma = _F.match(s_dom)
if ma:
try:
c = check.xreplace(ma)
except TypeError:
continue
if c == S.true:
debug('_inverse_laplace_apply_simple_rules match:')
debugf(' f: %s', (f,))
debugf(' rule: %s o---o %s', (s_dom, t_dom))
debugf(' ma: %s', (ma,))
return Heaviside(t)*t_dom.xreplace(ma).subs({t_: t}), S.true
return None
def _inverse_laplace_time_shift(F, s, t, plane):
"""
Helper function for the class InverseLaplaceTransform.
"""
a = Wild('a', exclude=[s])
g = Wild('g')
if not F.has(s):
return F*DiracDelta(t), S.true
ma1 = F.match(exp(a*s))
if ma1:
if ma1[a].is_negative:
debug('_inverse_laplace_time_shift match:')
debugf(' f: %s', (F,))
debug(' rule: exp(-a*s) o---o DiracDelta(t-a)')
debugf(' ma: %s', (ma1,))
return DiracDelta(t+ma1[a]), S.true
else:
debug('_inverse_laplace_time_shift match: negative time shift')
return InverseLaplaceTransform(F, s, t, plane), S.true
ma1 = F.match(exp(a*s)*g)
if ma1:
if ma1[a].is_negative:
debug('_inverse_laplace_time_shift match:')
debugf(' f: %s', (F,))
debug(' rule: exp(-a*s)*F(s) o---o Heaviside(t-a)*f(t-a)')
debugf(' ma: %s', (ma1,))
return _inverse_laplace_transform(ma1[g], s, t+ma1[a], plane)
else:
debug('_inverse_laplace_time_shift match: negative time shift')
return InverseLaplaceTransform(F, s, t, plane), S.true
return None
def _inverse_laplace_time_diff(F, s, t, plane):
"""
Helper function for the class InverseLaplaceTransform.
"""
n = Wild('n', exclude=[s])
g = Wild('g')
ma1 = F.match(s**n*g)
if ma1 and ma1[n].is_integer and ma1[n].is_positive:
debug('_inverse_laplace_time_diff match:')
debugf(' f: %s', (F,))
debug(' rule: s**n*F(s) o---o diff(f(t), t, n)')
debugf(' ma: %s', (ma1,))
r, c = _inverse_laplace_transform(ma1[g], s, t, plane)
r = r.replace(Heaviside(t), 1)
if r.has(InverseLaplaceTransform):
return diff(r, t, ma1[n]), c
else:
return Heaviside(t)*diff(r, t, ma1[n]), c
return None
def _inverse_laplace_apply_prog_rules(F, s, t, plane):
"""
Helper function for the class InverseLaplaceTransform.
"""
prog_rules = [_inverse_laplace_time_shift,
_inverse_laplace_time_diff]
for p_rule in prog_rules:
if (r := p_rule(F, s, t, plane)) is not None:
return r
return None
def _inverse_laplace_expand(fn, s, t, plane):
"""
Helper function for the class InverseLaplaceTransform.
"""
if fn.is_Add:
return None
r = expand(fn, deep=False)
if r.is_Add:
return _inverse_laplace_transform(r, s, t, plane)
r = expand_mul(fn)
if r.is_Add:
return _inverse_laplace_transform(r, s, t, plane)
r = expand(fn)
if r.is_Add:
return _inverse_laplace_transform(r, s, t, plane)
if fn.is_rational_function(s):
r = fn.apart(s).doit()
if r.is_Add:
return _inverse_laplace_transform(r, s, t, plane)
return None
def _inverse_laplace_rational(fn, s, t, plane, simplify):
"""
Helper function for the class InverseLaplaceTransform.
"""
debugf('[ILT _i_l_r] (%s, %s, %s)', (fn, s, t))
x_ = symbols('x_')
f = fn.apart(s)
terms = Add.make_args(f)
terms_t = []
conditions = [S.true]
for term in terms:
[n, d] = term.as_numer_denom()
dc = d.as_poly(s).all_coeffs()
dc_lead = dc[0]
dc = [x/dc_lead for x in dc]
nc = [x/dc_lead for x in n.as_poly(s).all_coeffs()]
if len(dc) == 1:
r = nc[0]*DiracDelta(t)
terms_t.append(r)
elif len(dc) == 2:
r = nc[0]*exp(-dc[1]*t)
terms_t.append(Heaviside(t)*r)
elif len(dc) == 3:
a = dc[1]/2
b = (dc[2]-a**2).factor()
if len(nc) == 1:
nc = [S.Zero] + nc
l, m = tuple(nc)
if b == 0:
r = (m*t+l*(1-a*t))*exp(-a*t)
else:
hyp = False
if b.is_negative:
b = -b
hyp = True
b2 = list(roots(x_**2-b, x_).keys())[0]
bs = sqrt(b).simplify()
if hyp:
r = (
l*exp(-a*t)*cosh(b2*t) + (m-a*l) /
bs*exp(-a*t)*sinh(bs*t))
else:
r = l*exp(-a*t)*cos(b2*t) + (m-a*l)/bs*exp(-a*t)*sin(bs*t)
terms_t.append(Heaviside(t)*r)
else:
ft, cond = _inverse_laplace_transform(
fn, s, t, plane, simplify=True, dorational=False)
terms_t.append(ft)
conditions.append(cond)
result = Add(*terms_t)
if simplify:
result = result.simplify(doit=False)
debugf('[ILT _i_l_r] returns %s', (result,))
return result, And(*conditions)
def _inverse_laplace_transform(
fn, s_, t_, plane, simplify=True, dorational=True):
"""
Front-end function of the inverse Laplace transform. It tries to apply all
known rules recursively. If everything else fails, it tries to integrate.
"""
terms = Add.make_args(fn)
terms_t = []
conditions = []
debugf('[ILT _i_l_t] (%s, %s, %s)', (fn, s_, t_))
for term in terms:
k, f = term.as_independent(s_, as_Add=False)
if (
dorational and term.is_rational_function(s_) and
(
r := _inverse_laplace_rational(
f, s_, t_, plane, simplify)) is not None):
pass
elif (r := _inverse_laplace_apply_simple_rules(f, s_, t_)) is not None:
pass
elif (r := _inverse_laplace_expand(f, s_, t_, plane)) is not None:
pass
elif (
(r := _inverse_laplace_apply_prog_rules(f, s_, t_, plane))
is not None):
pass
elif any(undef.has(s_) for undef in f.atoms(AppliedUndef)):
# If there are undefined functions f(t) then integration is
# unlikely to do anything useful so we skip it and given an
# unevaluated LaplaceTransform.
r = (InverseLaplaceTransform(f, s_, t_, plane), S.true)
elif (
r := _inverse_laplace_transform_integration(
f, s_, t_, plane, simplify=simplify)) is not None:
pass
else:
r = (InverseLaplaceTransform(f, s_, t_, plane), S.true)
(ri_, ci_) = r
terms_t.append(k*ri_)
conditions.append(ci_)
result = Add(*terms_t)
if simplify:
result = result.simplify(doit=False)
condition = And(*conditions)
return result, condition
class InverseLaplaceTransform(IntegralTransform):
"""
Class representing unevaluated inverse Laplace transforms.
For usage of this class, see the :class:`IntegralTransform` docstring.
For how to compute inverse Laplace transforms, see the
:func:`inverse_laplace_transform` docstring.
"""
_name = 'Inverse Laplace'
_none_sentinel = Dummy('None')
_c = Dummy('c')
def __new__(cls, F, s, x, plane, **opts):
if plane is None:
plane = InverseLaplaceTransform._none_sentinel
return IntegralTransform.__new__(cls, F, s, x, plane, **opts)
@property
def fundamental_plane(self):
plane = self.args[3]
if plane is InverseLaplaceTransform._none_sentinel:
plane = None
return plane
def _compute_transform(self, F, s, t, **hints):
return _inverse_laplace_transform_integration(
F, s, t, self.fundamental_plane, **hints)
def _as_integral(self, F, s, t):
c = self.__class__._c
return (
Integral(exp(s*t)*F, (s, c - S.ImaginaryUnit*S.Infinity,
c + S.ImaginaryUnit*S.Infinity)) /
(2*S.Pi*S.ImaginaryUnit))
def doit(self, **hints):
"""
Try to evaluate the transform in closed form.
Explanation
===========
Standard hints are the following:
- ``noconds``: if True, do not return convergence conditions. The
default setting is `True`.
- ``simplify``: if True, it simplifies the final result. The
default setting is `False`.
"""
_noconds = hints.get('noconds', True)
_simplify = hints.get('simplify', False)
debugf('[ILT doit] (%s, %s, %s)', (self.function,
self.function_variable,
self.transform_variable))
s_ = self.function_variable
t_ = self.transform_variable
fn = self.function
plane = self.fundamental_plane
r = _inverse_laplace_transform(fn, s_, t_, plane, simplify=_simplify)
if _noconds:
return r[0]
else:
return r
def inverse_laplace_transform(F, s, t, plane=None, **hints):
r"""
Compute the inverse Laplace transform of `F(s)`, defined as
.. math ::
f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} e^{st}
F(s) \mathrm{d}s,
for `c` so large that `F(s)` has no singularites in the
half-plane `\operatorname{Re}(s) > c-\epsilon`.
Explanation
===========
The plane can be specified by
argument ``plane``, but will be inferred if passed as None.
Under certain regularity conditions, this recovers `f(t)` from its
Laplace Transform `F(s)`, for non-negative `t`, and vice
versa.
If the integral cannot be computed in closed form, this function returns
an unevaluated :class:`InverseLaplaceTransform` object.
Note that this function will always assume `t` to be real,
regardless of the SymPy assumption on `t`.
For a description of possible hints, refer to the docstring of
:func:`sympy.integrals.transforms.IntegralTransform.doit`.
Examples
========
>>> from sympy import inverse_laplace_transform, exp, Symbol
>>> from sympy.abc import s, t
>>> a = Symbol('a', positive=True)
>>> inverse_laplace_transform(exp(-a*s)/s, s, t)
Heaviside(-a + t)
See Also
========
laplace_transform
hankel_transform, inverse_hankel_transform
"""
if isinstance(F, MatrixBase) and hasattr(F, 'applyfunc'):
return F.applyfunc(
lambda Fij: inverse_laplace_transform(Fij, s, t, plane, **hints))
return InverseLaplaceTransform(F, s, t, plane).doit(**hints)
def _fast_inverse_laplace(e, s, t):
"""Fast inverse Laplace transform of rational function including RootSum"""
a, b, n = symbols('a, b, n', cls=Wild, exclude=[s])
def _ilt(e):
if not e.has(s):
return e
elif e.is_Add:
return _ilt_add(e)
elif e.is_Mul:
return _ilt_mul(e)
elif e.is_Pow:
return _ilt_pow(e)
elif isinstance(e, RootSum):
return _ilt_rootsum(e)
else:
raise NotImplementedError
def _ilt_add(e):
return e.func(*map(_ilt, e.args))
def _ilt_mul(e):
coeff, expr = e.as_independent(s)
if expr.is_Mul:
raise NotImplementedError
return coeff * _ilt(expr)
def _ilt_pow(e):
match = e.match((a*s + b)**n)
if match is not None:
nm, am, bm = match[n], match[a], match[b]
if nm.is_Integer and nm < 0:
return t**(-nm-1)*exp(-(bm/am)*t)/(am**-nm*gamma(-nm))
if nm == 1:
return exp(-(bm/am)*t) / am
raise NotImplementedError
def _ilt_rootsum(e):
expr = e.fun.expr
[variable] = e.fun.variables
return RootSum(e.poly, Lambda(variable, together(_ilt(expr))))
return _ilt(e)