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src/sage/functions/exp_integral.py: More block tags
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Matthias Koeppe committed Aug 20, 2023
1 parent 59c63ce commit 794a1eb
Showing 1 changed file with 102 additions and 100 deletions.
202 changes: 102 additions & 100 deletions src/sage/functions/exp_integral.py
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Original file line number Diff line number Diff line change
Expand Up @@ -116,16 +116,15 @@ class Function_exp_integral_e(BuiltinFunction):
Symbolic derivatives and integrals are handled by Sage and Maxima::
sage: x = var('x') # needs sage.symbolic
sage: f = exp_integral_e(2,x) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: # needs sage.symbolic
sage: x = var('x')
sage: f = exp_integral_e(2,x)
sage: f.diff(x)
-exp_integral_e(1, x)
sage: f.integrate(x) # needs sage.symbolic
sage: f.integrate(x)
-exp_integral_e(3, x)
sage: f = exp_integral_e(-1, x) # needs sage.symbolic
sage: f.integrate(x) # needs sage.symbolic
sage: f = exp_integral_e(-1, x)
sage: f.integrate(x)
Ei(-x) - gamma(-1, x)
Some special values of ``exp_integral_e`` can be simplified.
Expand Down Expand Up @@ -298,12 +297,12 @@ class Function_exp_integral_e1(BuiltinFunction):
Symbolic derivatives and integrals are handled by Sage and Maxima::
sage: x = var('x') # needs sage.symbolic
sage: f = exp_integral_e1(x) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: # needs sage.symbolic
sage: x = var('x')
sage: f = exp_integral_e1(x)
sage: f.diff(x)
-e^(-x)/x
sage: f.integrate(x) # needs sage.symbolic
sage: f.integrate(x)
-exp_integral_e(2, x)
ALGORITHM:
Expand Down Expand Up @@ -359,13 +358,13 @@ def _derivative_(self, z, diff_param=None):
EXAMPLES::
sage: x = var('x') # needs sage.symbolic
sage: f = exp_integral_e1(x) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: # needs sage.symbolic
sage: x = var('x')
sage: f = exp_integral_e1(x)
sage: f.diff(x)
-e^(-x)/x
sage: f = exp_integral_e1(x^2) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: f = exp_integral_e1(x^2)
sage: f.diff(x)
-2*e^(-x^2)/x
"""
Expand Down Expand Up @@ -400,19 +399,19 @@ class Function_log_integral(BuiltinFunction):
Symbolic derivatives and integrals are handled by Sage and Maxima::
sage: x = var('x') # needs sage.symbolic
sage: f = log_integral(x) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: # needs sage.symbolic
sage: x = var('x')
sage: f = log_integral(x)
sage: f.diff(x)
1/log(x)
sage: f.integrate(x) # needs sage.symbolic
sage: f.integrate(x)
x*log_integral(x) - Ei(2*log(x))
Here is a test from the mpmath documentation. There are
1,925,320,391,606,803,968,923 many prime numbers less than 1e23. The
value of ``log_integral(1e23)`` is very close to this::
sage: log_integral(1e23) # needs sage.symbolic
sage: log_integral(1e23) # needs mpmath
1.92532039161405e21
ALGORITHM:
Expand Down Expand Up @@ -479,7 +478,7 @@ def _evalf_(self, z, parent=None, algorithm=None):
"""
EXAMPLES::
sage: N(log_integral(1e6)) # needs sage.symbolic
sage: N(log_integral(1e6)) # needs mpmath
78627.5491594622
sage: log_integral(RealField(200)(1e6)) # needs sage.rings.real_mpfr
78627.549159462181919862910747947261161321874382421767074759
Expand All @@ -493,13 +492,13 @@ def _derivative_(self, z, diff_param=None):
EXAMPLES::
sage: x = var('x') # needs sage.symbolic
sage: f = log_integral(x) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: # needs sage.symbolic
sage: x = var('x')
sage: f = log_integral(x)
sage: f.diff(x)
1/log(x)
sage: f = log_integral(x^2) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: f = log_integral(x^2)
sage: f.diff(x)
2*x/log(x^2)
"""
Expand Down Expand Up @@ -584,7 +583,7 @@ class Function_log_integral_offset(BuiltinFunction):
1.2428254968641898308632562019 + 0.87232935488528370139883806779*I
sage: log_integral_offset(2)
0
sage: for n in range(1,7):
sage: for n in range(1,7): # needs primecountpy
....: print('%-10s%-10s%-20s'%(10^n, prime_pi(10^n), N(Li(10^n))))
10 4 5.12043572466980
100 25 29.0809778039621
Expand Down Expand Up @@ -689,13 +688,13 @@ def _derivative_(self, z, diff_param=None):
EXAMPLES::
sage: x = var('x') # needs sage.symbolic
sage: f = log_integral_offset(x) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: # needs sage.symbolic
sage: x = var('x')
sage: f = log_integral_offset(x)
sage: f.diff(x)
1/log(x)
sage: f = log_integral_offset(x^2) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: f = log_integral_offset(x^2)
sage: f.diff(x)
2*x/log(x^2)
"""
return 1/log(z)
Expand Down Expand Up @@ -736,7 +735,7 @@ class Function_sin_integral(BuiltinFunction):
The limit of `\operatorname{Si}(z)` as `z \to \infty` is `\pi/2`::
sage: N(sin_integral(1e23)) # needs sage.symbolic
sage: N(sin_integral(1e23)) # needs mpmath
1.57079632679490
sage: N(pi/2) # needs sage.symbolic
1.57079632679490
Expand All @@ -760,15 +759,14 @@ class Function_sin_integral(BuiltinFunction):
Symbolic derivatives and integrals are handled by Sage and Maxima::
sage: x = var('x') # needs sage.symbolic
sage: f = sin_integral(x) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: # needs sage.symbolic
sage: x = var('x')
sage: f = sin_integral(x)
sage: f.diff(x)
sin(x)/x
sage: f.integrate(x) # needs sage.symbolic
sage: f.integrate(x)
x*sin_integral(x) + cos(x)
sage: integrate(sin(x)/x, x) # needs sage.symbolic
sage: integrate(sin(x)/x, x)
-1/2*I*Ei(I*x) + 1/2*I*Ei(-I*x)
Expand Down Expand Up @@ -883,13 +881,13 @@ def _derivative_(self, z, diff_param=None):
EXAMPLES::
sage: x = var('x') # needs sage.symbolic
sage: f = sin_integral(x) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: # needs sage.symbolic
sage: x = var('x')
sage: f = sin_integral(x)
sage: f.diff(x)
sin(x)/x
sage: f = sin_integral(x^2) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: f = sin_integral(x^2)
sage: f.diff(x)
2*sin(x^2)/x
"""
Expand Down Expand Up @@ -919,7 +917,7 @@ class Function_cos_integral(BuiltinFunction):
0.119629786008000
sage: cos_integral(0) # needs sage.symbolic
cos_integral(0)
sage: N(cos_integral(0)) # needs sage.symbolic
sage: N(cos_integral(0)) # needs mpmath
-infinity
Numerical evaluation for real and complex arguments is handled using mpmath::
Expand Down Expand Up @@ -953,12 +951,12 @@ class Function_cos_integral(BuiltinFunction):
Symbolic derivatives and integrals are handled by Sage and Maxima::
sage: x = var('x') # needs sage.symbolic
sage: f = cos_integral(x) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: # needs sage.symbolic
sage: x = var('x')
sage: f = cos_integral(x)
sage: f.diff(x)
cos(x)/x
sage: f.integrate(x) # needs sage.symbolic
sage: f.integrate(x)
x*cos_integral(x) - sin(x)
The Nielsen spiral is the parametric plot of (Si(t), Ci(t))::
Expand Down Expand Up @@ -1025,13 +1023,13 @@ def _derivative_(self, z, diff_param=None):
EXAMPLES::
sage: x = var('x') # needs sage.symbolic
sage: f = cos_integral(x) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: # needs sage.symbolic
sage: x = var('x')
sage: f = cos_integral(x)
sage: f.diff(x)
cos(x)/x
sage: f = cos_integral(x^2) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: f = cos_integral(x^2)
sage: f.diff(x)
2*cos(x^2)/x
"""
Expand Down Expand Up @@ -1083,7 +1081,7 @@ class Function_sinh_integral(BuiltinFunction):
The limit `\operatorname{Shi}(z)` as `z \to \infty` is `\infty`::
sage: N(sinh_integral(Infinity)) # needs sage.symbolic
sage: N(sinh_integral(Infinity)) # needs mpmath
+infinity
Symbolic derivatives and integrals are handled by Sage and Maxima::
Expand All @@ -1103,9 +1101,9 @@ class Function_sinh_integral(BuiltinFunction):
sage: integrate(sinh_integral(x), x, 0, 1/2) # needs sage.symbolic
-cosh(1/2) + 1/2*sinh_integral(1/2) + 1
sage: integrate(sinh_integral(x), x, 0, 1/2).n() # correct # needs sage.symbolic
sage: integrate(sinh_integral(x), x, 0, 1/2).n() # correct # needs sage.symbolic
0.125872409703453
sage: integrate(sinh_integral(x), x, 0, 0.5).n() # fixed in maxima 5.29.1 # needs sage.symbolic
sage: integrate(sinh_integral(x), x, 0, 0.5).n() # fixed in maxima 5.29.1 # needs sage.symbolic
0.125872409703453
ALGORITHM:
Expand Down Expand Up @@ -1177,13 +1175,13 @@ def _derivative_(self, z, diff_param=None):
EXAMPLES::
sage: x = var('x') # needs sage.symbolic
sage: f = sinh_integral(x) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: # needs sage.symbolic
sage: x = var('x')
sage: f = sinh_integral(x)
sage: f.diff(x)
sinh(x)/x
sage: f = sinh_integral(ln(x)) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: f = sinh_integral(ln(x))
sage: f.diff(x)
1/2*(x^2 - 1)/(x^2*log(x))
"""
Expand Down Expand Up @@ -1224,7 +1222,7 @@ class Function_cosh_integral(BuiltinFunction):
Here is an example from the mpmath documentation::
sage: f(x) = cosh_integral(x) # needs sage.symbolic
sage: find_root(f, 0.1, 1.0) # needs sage.symbolic
sage: find_root(f, 0.1, 1.0) # needs scipy sage.symbolic
0.523822571389...
Compare ``cosh_integral(3.0)`` to the definition of the value using
Expand All @@ -1243,17 +1241,17 @@ class Function_cosh_integral(BuiltinFunction):
The limit of `\operatorname{Chi}(z)` as `z \to \infty` is `\infty`::
sage: N(cosh_integral(Infinity)) # needs sage.symbolic
sage: N(cosh_integral(Infinity)) # needs mpmath
+infinity
Symbolic derivatives and integrals are handled by Sage and Maxima::
sage: x = var('x') # needs sage.symbolic
sage: f = cosh_integral(x) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: # needs sage.symbolic
sage: x = var('x')
sage: f = cosh_integral(x)
sage: f.diff(x)
cosh(x)/x
sage: f.integrate(x) # needs sage.symbolic
sage: f.integrate(x)
x*cosh_integral(x) - sinh(x)
ALGORITHM:
Expand Down Expand Up @@ -1305,13 +1303,13 @@ def _derivative_(self, z, diff_param=None):
EXAMPLES::
sage: x = var('x') # needs sage.symbolic
sage: f = cosh_integral(x) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: # needs sage.symbolic
sage: x = var('x')
sage: f = cosh_integral(x)
sage: f.diff(x)
cosh(x)/x
sage: f = cosh_integral(ln(x)) # needs sage.symbolic
sage: f.diff(x) # needs sage.symbolic
sage: f = cosh_integral(ln(x))
sage: f.diff(x)
1/2*(x^2 + 1)/(x^2*log(x))
"""
Expand Down Expand Up @@ -1344,16 +1342,18 @@ class Function_exp_integral(BuiltinFunction):
EXAMPLES::
sage: Ei(10) # needs sage.symbolic
sage: # needs sage.symbolic
sage: Ei(10)
Ei(10)
sage: Ei(I) # needs sage.symbolic
sage: Ei(I)
Ei(I)
sage: Ei(3+I) # needs sage.symbolic
sage: Ei(3+I)
Ei(I + 3)
sage: Ei(10r)
Ei(10)
sage: Ei(1.3) # needs mpmath
2.72139888023202
sage: Ei(10r) # needs sage.symbolic
Ei(10)
sage: Ei(1.3r) # needs mpmath
2.7213988802320235
Expand Down Expand Up @@ -1470,22 +1470,24 @@ def exponential_integral_1(x, n=0):
EXAMPLES::
sage: exponential_integral_1(2) # needs sage.libs.pari
sage: # needs sage.libs.pari
sage: exponential_integral_1(2)
0.0489005107080611
sage: exponential_integral_1(2, 4) # abs tol 1e-18 # needs sage.libs.pari
sage: exponential_integral_1(2, 4) # abs tol 1e-18
[0.0489005107080611, 0.00377935240984891, 0.000360082452162659, 0.0000376656228439245]
sage: exponential_integral_1(40, 5) # needs sage.libs.pari
sage: exponential_integral_1(40, 5)
[0.000000000000000, 2.22854325868847e-37, 6.33732515501151e-55,
2.02336191509997e-72, 6.88522610630764e-90]
sage: exponential_integral_1(0)
+Infinity
sage: r = exponential_integral_1(RealField(150)(1)); r # needs sage.libs.pari
sage: r = exponential_integral_1(RealField(150)(1)); r
0.21938393439552027367716377546012164903104729
sage: parent(r) # needs sage.libs.pari
sage: parent(r)
Real Field with 150 bits of precision
sage: exponential_integral_1(RealField(150)(100)) # needs sage.libs.pari
sage: exponential_integral_1(RealField(150)(100))
3.6835977616820321802351926205081189876552201e-46
sage: exponential_integral_1(0)
+Infinity
TESTS:
The relative error for a single value should be less than 1 ulp::
Expand Down

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