ASKSAGE: Sage Q&A Forum - Latest question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 24 Dec 2019 23:19:43 -0600why sagemath can integrate a string?https://ask.sagemath.org/question/49165/why-sagemath-can-integrate-a-string/ I am surprised this works. Sagemath 8.9
sage: var('x')
x
sage: integrate("x",x)
1/2*x^2
sage: type("x")
<type 'str'>
Why does sagemath accept string for the integrand? Should not this be type error? Maple:
restart
int("x",x)
Error, (in int) wrong number (or type) of arguments: wrong type of integrand passed to indefinite integration.
NasserTue, 24 Dec 2019 07:46:54 -0600https://ask.sagemath.org/question/49165/Unable to parse Giac output errorhttps://ask.sagemath.org/question/49176/unable-to-parse-giac-output-error/Sagemath 8.9
Why does sagemath return this error here?
sage: var('x')
x
sage: integrate((1-2*x^(1/3))^(3/4)/x,x, algorithm="giac")
NotImplementedError Traceback (most recent call last)
<ipython-input-16-987ddabbc645> in <module>()
----> 1 integrate((Integer(1)-Integer(2)*x**(Integer(1)/Integer(3)))**(Integer(3)/Integer(4))/x,x, algorithm="giac")
/usr/lib/python2.7/site-packages/sage/misc/functional.pyc in integral(x, *args, **kwds)
751 """
752 if hasattr(x, 'integral'):
--> 753 return x.integral(*args, **kwds)
754 else:
755 from sage.symbolic.ring import SR
/usr/lib/python2.7/site-packages/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression.integral (build/cythonized/sage/symbolic/expression.cpp:64032)()
12360 R = ring.SR
12361 return R(integral(f, v, a, b, **kwds))
> 12362 return integral(self, *args, **kwds)
12363
12364 integrate = integral
/usr/lib/python2.7/site-packages/sage/symbolic/integration/integral.pyc in integrate(expression, v, a, b, algorithm, hold)
910 if not integrator:
911 raise ValueError("Unknown algorithm: %s" % algorithm)
--> 912 return integrator(expression, v, a, b)
913 if a is None:
914 return indefinite_integral(expression, v, hold=hold)
/usr/lib/python2.7/site-packages/sage/symbolic/integration/external.pyc in giac_integrator(expression, v, a, b)
430 return expression.integrate(v, a, b, hold=True)
431 else:
--> 432 return result._sage_()
/usr/lib/python2.7/site-packages/sage/interfaces/giac.pyc in _sage_(self, locals)
1096
1097 except Exception:
-> 1098 raise NotImplementedError("Unable to parse Giac output: %s" % result)
1099 else:
1100 return [entry.sage() for entry in self]
NotImplementedError: Unable to parse Giac output: Evaluation time: 1.76
12*(1/4*ln(abs((-2*x^(1/3)+1)^(1/4)-1))-1/4*ln((-2*x^(1/3)+1)^(1/4)+1)+1/2*atan((-2*x^(1/3)+1)^(1/4))+1/3*((-2*x^(1/3)+1)^(1/4))^3)
sage:
The error is similar to one in this bug report from 3 years ago [https://trac.sagemath.org/ticket/22997](https://trac.sagemath.org/ticket/22997) but that is for unresolved integral while here Giac is able to solve it.
Here is the same thing using giac directly on same computer
>giac
// Using locale /usr/share/locale/
// en_US.utf8
// /usr/share/locale/
// giac
// UTF-8
// Maximum number of parallel threads 4
Help file /usr/share/giac/doc/en/aide_cas not found
Added 26 synonyms
Welcome to giac readline interface
(c) 2001,2018 B. Parisse & others
Homepage http://www-fourier.ujf-grenoble.fr/~parisse/giac.html
Released under the GPL license 3.0 or above
See http://www.gnu.org for license details
May contain BSD licensed software parts (lapack, atlas, tinymt)
-------------------------------------------------
Press CTRL and D simultaneously to finish session
Type ?commandname for help
0>> integrate((1-2*x^(1/3))^(3/4)/x,x)
Evaluation time: 1.66
12*(1/4*ln(abs((-2*x^(1/3)+1)^(1/4)-1))-1/4*ln((-2*x^(1/3)+1)^(1/4)+1)+1/2*atan((-2*x^(1/3)+1)^(1/4))+1/3*((-2*x^(1/3)+1)^(1/4))^3)
// Time 1.66
1>>
Version
>giac --version
// Using locale /usr/share/locale/
// en_US.utf8
// /usr/share/locale/
// giac
// UTF-8
// Maximum number of parallel threads 4
// (c) 2001, 2018 B. Parisse & others
1.5.0
>
Any suggestions what is going on?
Thanks
--Nasser
NasserTue, 24 Dec 2019 23:19:43 -0600https://ask.sagemath.org/question/49176/Solved: Why does integrate(psi(y)*f(y),y) return an error but integrate(psi(t,y)*f(t,y),y) works?https://ask.sagemath.org/question/43287/solved-why-does-integratepsiyfyy-return-an-error-but-integratepsityftyy-works/Hi there,
I am trying get an symbolic expression for the convolution
$$ (\psi \star f)(x) := \int\limits_{\mathbb{R}} \psi(x-y) f(y) {d y} $$
of two functions
$
f, \psi: \mathbb{R} \to \mathbb{R}
$
as follows:
<code>
var('y') <br>
psi = function('psi')(y) <br>
f = function('f')(y) <br>
integrate(psi(x-y)*f(y),y)
</code>
upon which I get the error message
> RuntimeError: ECL says: Error executing code in Maxima:
If I add an extra argument to the two functions and define them as
$$ f, \psi : \mathbb{R} \times \mathbb{R} \to \mathbb{R} $$
as follows:
<code>
var('t') <br>
psi = function('psi')(t,y) <br>
f = function('f')(t,y) <br>
integrate(psi(t,x-y)*f(t,y),y)
</code>
there is a surprise, *it suddenly works!*
I get the desired symbolic expression on which I can run diff(..,x) and all the other built-in functions.
**TL;DR**
Why does <code>integrate(psi(y)*f(y),y)</code> return an error?
**Solution**
Use sympy backend for symbolic integration as in
<code>integrate(psi(x-y)*f(y),y, algorithm="sympy")</code>hausdorffWed, 08 Aug 2018 06:41:12 -0500https://ask.sagemath.org/question/43287/Analytical evaluation of Fermi-Dirac integralshttps://ask.sagemath.org/question/45559/analytical-evaluation-of-fermi-dirac-integrals/It seems that sagemath is unable to calculate Fermic-Dirac type integrals, e. g.
integrate(x^2/(1+ e^x),x,0,oo) =>
limit(1/3*x^3 - x^2*log(e^x + 1) - 2*x*dilog(-e^x) + 2*polylog(3, -e^x), x, +Infinity, minus) + 3/2*zeta(3)
integrate(x^3/(1+ e^x),x,0,oo)==>
-7/120*pi^4 + limit(1/4*x^4 - x^3*log(e^x + 1) - 3*x^2*dilog(-e^x) + 6*x*polylog(3, -e^x) - 6*polylog(4, -e^x), x,
+Infinity, minus)
Normally, the former evaluates to (3*Zeta[3])/2 and the latter to (7*Pi^4)/120, using Mathematica.irizosWed, 27 Feb 2019 07:55:41 -0600https://ask.sagemath.org/question/45559/Sage could be even more clever - How to force the use of 'sympy' backend for simplifying symbolic integrals?https://ask.sagemath.org/question/43392/sage-could-be-even-more-clever-how-to-force-the-use-of-sympy-backend-for-simplifying-symbolic-integrals/Hi there,
I have noticed the following problem:
sage: f = function('f')(x)
sage: var('h')
sage: integrate(exp(h)*exp(x)*f(x),x)
integrate(e^(h + x)*f(x), x)
The workaround seems to be using the `sympy` backend for symbolic integration
sage: integrate(exp(h)*exp(x)*f(x),x,algorithm='sympy')
e^h*integrate(e^x*f(x), x)
which always seems to be a good idea as I learned from @Emmanuel Charpentier over
[here](https://ask.sagemath.org/question/43287/solved-why-does-integratepsiyfyy-return-an-error-but-integratepsityftyy-works/?answer=43297#post-id-43297).
Now I would like to force the use of `algorith='sympy'` for simplifying these `integrate(...)` expressions globally.
Unfortunately, the `simplify()` command does not allow to set this option.
sage: integrate(exp(h)*exp(x)*f(x),x)
integrate(e^(h + x)*f(x), x)
sage: _.simplify()
integrate(e^(h + x)*f(x), x)
**TL;DR** How can I force sage to pull out these type of exponential constants from the integral with the `simplify()` command?
hausdorffThu, 16 Aug 2018 11:54:20 -0500https://ask.sagemath.org/question/43392/Integrate expression involving a formal functionhttps://ask.sagemath.org/question/41415/integrate-expression-involving-a-formal-function/I am trying to integrate expressions involving a formal function $f(x)$:
sage: var('x')
sage: function('f')
It works in some simple cases:
sage: integrate(cos(f(x)) * f(x).diff(), x)
sin(f(x))
However, a slightly more complicated expression is left unchanged:
sage: integrate(x * f(x).diff() + f(x), x)
integrate(x*diff(f(x), x) + f(x), x)
instead of simplifying to $x\ f(x)$.
Is it possible to integrate such expressions in Sage? I tried to use other algorithms (e.g. `algorithm='mathematica_free'`), but I got this traceback:
.local/opt/sage-8.0/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.pyc in derivative(self, ex, operator)
588 from sage.symbolic.ring import is_SymbolicVariable
589 if self.name_init != "_maxima_init_":
--> 590 raise NotImplementedError
591 args = ex.operands()
592 if (not all(is_SymbolicVariable(v) for v in args) or
so it looks like only Maxima algorithm can be used because the derivative operation cannot be converted to anything else.envypoleTue, 06 Mar 2018 12:41:49 -0600https://ask.sagemath.org/question/41415/Difference between integral(csc(x)) and integral(1/sin(x))?https://ask.sagemath.org/question/40806/difference-between-integralcscx-and-integral1sinx/ integral(csc(x),x) gives -log(cot(x) + csc(x)) as expected. integral(1/sin(x),x) gives -1/2*log(cos(x) + 1) + 1/2*log(cos(x) - 1). Evaluation of the second log is problematic because cos(x)-1 < 0 for all x except n*2pi. Choosing the sympy or maxima algorithms makes no difference.
Also (and this is less reliable) if I perform a substitution t=tan(x/2) by hand in the first case I get ln(t), again as expected. But in the second ln(-t). That maybe my fault but I can't see what I did wrong.
Is there any fundamental difference between asking that same question in two different ways?
NickBaileySat, 27 Jan 2018 09:10:05 -0600https://ask.sagemath.org/question/40806/integrate x^3/(exp(x)-1) between 0 and infinityhttps://ask.sagemath.org/question/37865/integrate-x3expx-1-between-0-and-infinity/If I type
> integrate(x^3/(exp(x)-1),x,0,infinity)
I get
-1/15*pi^4 + limit(-1/4*x^4 + x^3*log(-e^x + 1) + 3*x^2*dilog(e^x) - 6*x*polylog(3, e^x) + 6*polylog(4, e^x), x, +Infinity, minus)
The command numerical_integral(x^3/(exp(x)-1),0,infinity) gives 6.4939394075
I have two questions :
1. How do I evaluate the limit ?
2. The correct answer is pi^4/15(=6.49393940226683) : why
SageMath does not give it with symbolic integration ?
ThanksepimetheusThu, 08 Jun 2017 12:56:33 -0500https://ask.sagemath.org/question/37865/symbolic integrationhttps://ask.sagemath.org/question/36408/symbolic-integration/ If you ask sage to symbolically integrate the following properly, the answer is wrong. Why?
cos(x)/(a*cos(x) + b*sin(x))
[Aside -- the captcha was invisible with firefox 50.1/linux, leading to much teeth-gnashing. Seems to work OK with Chrome. Not happy]bevSat, 28 Jan 2017 22:44:30 -0600https://ask.sagemath.org/question/36408/Two ways of integrating x↦xⁿsin(x) give contradictory results. Bug?https://ask.sagemath.org/question/36185/two-ways-of-integrating-x-xnsinx-give-contradictory-results-bug/**First way:**
var('x,n')
integral(x^n*sin(x),x)
gives just
integrate(x^n*sin(x), x)
not very informative, let us try to add an assumption to get nicer results.
**Second way:**
assume(n,'integer')
integral(x^n*sin(x),x)
gives
1/4*(((-1)^n - 1)*gamma(n + 1, I*x) - ((-1)^n - 1)*gamma(n + 1, -I*x))*(-1)^(-1/2*n)
Uhm, looks better, but... wait, isn't `(-1)^n-1` equal to `0` for even values of `n` ? That would make the whole thing equal to `0` for even `n`.
I = integral(x^n*sin(x),x)
for k in range(10):
print I.subs(n==2*k)
prints only `0`s. Weird, non-zero functions should not have zero integrals.
**Third way :**
Let us try to do the integration with particular values of `n`.
for n in range(5):
print integral(x^n*sin(x),x)
prints
-cos(x)
-x*cos(x) + sin(x)
-(x^2 - 2)*cos(x) + 2*x*sin(x)
-(x^3 - 6*x)*cos(x) + 3*(x^2 - 2)*sin(x)
Looks better, but is clearly different from the previous answer.
**Question:**
I am working on the cloud, with SageMath 7.4 kernel. Is this a bug or did I misunderstood the meaning of the `'integer'`assumption ?
If this is a bug, how should I report it, is posting this question here enough ?
P.S. I did read the [wiki page about reporting bugs](http://doc.sagemath.org/html/en/developer/trac.html#reporting-bugs), but, gosh, is it really necessary to have a google account in order to report a bug ? Both sage-devel and sage-support are on Google Groups. lbWed, 04 Jan 2017 14:44:16 -0600https://ask.sagemath.org/question/36185/integration of matrix-valued functionhttps://ask.sagemath.org/question/31085/integration-of-matrix-valued-function/ Hi,
I want to compute the (in)definite integral of matrix valued function, for example
var('x')
f =matrix([[x,1],[x^2,2]]);
The command
integrate(f,x,0,1)
doesn't work. I expect matrix as a result - integrated by elements
janThu, 26 Nov 2015 06:34:42 -0600https://ask.sagemath.org/question/31085/help with simple integration of piecewise function?https://ask.sagemath.org/question/24887/help-with-simple-integration-of-piecewise-function/ r = var('r')
Piecewise([[(1,2), 1/floor(r)]]).integral(r,1,2)
Gives an error:
Error in lines ...
AttributeError: 'sage.rings.integer.Integer' object has no attribute 'variables
What am I doing wrong? I note that
Piecewise([[(1,2), 1/floor(r)]]).integral(r)
gives output ``Piecewise defined function with 1 parts, [[(1, 2), r1 |--> integrate(1/floor(r1), r1, 1, r1)]]''.
And that
integrate(1/floor(r1), r1, 1, r1)(2)
gives the same error as the first attempt. Here's the full stacktrace of the error:
EDIT: Thanks. Here's the full stack trace for the error:
File "/Applications/Sage-6.3.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/functions/piecewise.py", line 833, in integral
return F(b) - F(a)
File "/Applications/Sage-6.3.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/functions/piecewise.py", line 665, in __call__
return self.functions()[n-1](x0)
File "expression.pyx", line 4391, in sage.symbolic.expression.Expression.__call__ (build/cythonized/sage/symbolic/expression.cpp:21933)
File "/Applications/Sage-6.3.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/symbolic/callable.py", line 477, in _call_element_
return SR(_the_element.substitute(**d))
File "expression.pyx", line 4242, in sage.symbolic.expression.Expression.substitute (build/cythonized/sage/symbolic/expression.cpp:21183)
File "/Applications/Sage-6.3.app/Contents/Resources/sage/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.py", line 161, in _eval_
if len(x.variables()) == 1:
File "element.pyx", line 344, in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4022)
File "misc.pyx", line 257, in sage.structure.misc.getattr_from_other_class (build/cythonized/sage/structure/misc.c:1775)
AttributeError: 'sage.rings.integer.Integer' object has no attribute 'variables'NealSun, 16 Nov 2014 16:35:36 -0600https://ask.sagemath.org/question/24887/Working with formal power serieshttps://ask.sagemath.org/question/10553/working-with-formal-power-series/This is a simplified version of my previous [question](http://ask.sagemath.org/question/3012/integrating-formal-laurent-series).
1) Is it possible to define a formal power series in sage by giving an expression for the n-th coefficient, e.g. as the expression "n" defines the power series 0 + 1 x + 2 x^2 + 3 x^3 + ... n x^n + ... ?
2) Does sage know how to multiply such objects by convolving the terms? Can it anti-differentiate them symbolically?
anilbvFri, 20 Sep 2013 03:17:18 -0500https://ask.sagemath.org/question/10553/integrating formal Laurent serieshttps://ask.sagemath.org/question/10546/integrating-formal-laurent-series/I would like to compute some integrals of products of Laurent series, with the goal of getting an explicit expression for the n-th coefficient of the result. It gets pretty messy so I was hoping that sage could keep track of the details for me. Here are the things I don't know how to do:
1) Define a Laurent series by giving an expression for its n-th coefficient.
2) Formally multiply and integrate Laurent series, producing an explicit expression for the n-th coefficient of the result.
Is this possible? I apologize if some or all of this is explained elsewhere.
EDIT: An example of what I want to do would be to define a power series such as sum(n * x^n,n,0,infinity) and then integrate it and/or multiply it by another power series, resulting in yet another power series whose coefficients I can read off. The O(x^n) notation is not very useful to me since I would like to have an expression for an arbitrary coefficient of that resulting series in terms of n.anilbvWed, 18 Sep 2013 12:39:16 -0500https://ask.sagemath.org/question/10546/integrate cos(x)*cos(2x)*...*cos(mx) via SAGEhttps://ask.sagemath.org/question/10248/integrate-cosxcos2xcosmx-via-sage/I'm going to find $I_m=\int_0^{2\pi} \prod_{k=1}^m cos(kx)\{}dx$, where $m=1,2,3\ldots$
Simple SAGE code:
x=var('x')
f = lambda m,x : prod([cos(k*x) for k in range(1,m+1)])
for m in range(1,15+1):
print m, numerical_integral(f(m,x), 0, 2*pi)[0],integrate(f(m,x),x,0,2*pi).n()
Output:
1 -1.47676658757e-16 0.000000000000000
2 -5.27735962315e-16 0.000000000000000
3 1.57079632679 1.57079632679490
4 0.785398163397 0.785398163397448
5 -2.60536121164e-16 0.000000000000000
6 -1.81559273097e-16 0.000000000000000
7 0.392699081699 0.392699081698724
8 0.343611696486 0.147262155637022
9 -1.72448482421e-16 0.294524311274043
10 -1.8747663502e-16 0.196349540849362
11 0.214757310304 0.312932080728671
12 0.190213617698 0.177941771394734
13 -1.30355375996e-16 0.208621387152447
14 -1.25168280013e-16 0.0859029241215959
15 0.138441766107 0.134223318939994
As you can see numerical answer is right, but result of integrate(...) is right for $m=1,2,\ldots,7$ and then there is some bug.
We can print indefinite integral:
for m in range(7,11+1):
print 'm=',m
print 'Indef_I_m=',integrate(f(m,x),x)
And Output:
m = 7
Indef_I_m = 1/16*x + 1/16*sin(2*x) + 1/32*sin(4*x) + 7/384*sin(6*x) +
7/512*sin(8*x) + 3/320*sin(10*x) + 5/768*sin(12*x) + 5/896*sin(14*x) +
1/256*sin(16*x) + 1/384*sin(18*x) + 1/640*sin(20*x) + 1/704*sin(22*x) +
1/1536*sin(24*x) + 1/1664*sin(26*x) + 1/1792*sin(28*x)
m = 8
Indef_I_m = 3/128*x + 5/256*sin(2*x) + 1/32*sin(3*x) + 5/512*sin(4*x) +
5/768*sin(6*x) + 1/256*sin(8*x) + 1/256*sin(10*x) + 1/256*sin(12*x) +
1/256*sin(14*x) + 1/256*sin(16*x) + 7/2304*sin(18*x) + 3/1280*sin(20*x)
+ 5/2816*sin(22*x) + 1/768*sin(24*x) + 3/3328*sin(26*x) +
1/1792*sin(28*x) + 1/1920*sin(30*x) + 1/4096*sin(32*x) +
1/4352*sin(34*x) + 1/4608*sin(36*x) + 3/32*sin(x)
m = 9
Indef_I_m = 3/64*x + 3/128*sin(2*x) + 23/768*sin(3*x) + 3/256*sin(4*x) +
3/640*sin(5*x) + 1/128*sin(6*x) + 5/1792*sin(7*x) + 5/2304*sin(9*x) +
3/2816*sin(11*x) + 1/832*sin(13*x) + 1/1280*sin(15*x) + 3/4352*sin(17*x)
+ 5/4864*sin(19*x) + 1/1344*sin(21*x) + 3/2944*sin(23*x) +
7/6400*sin(25*x) + 1/1152*sin(27*x) + 3/3712*sin(29*x) +
5/7936*sin(31*x) + 1/2112*sin(33*x) + 3/8960*sin(35*x) +
1/4736*sin(37*x) + 1/4992*sin(39*x) + 1/10496*sin(41*x) +
1/11008*sin(43*x) + 1/11520*sin(45*x) + 23/256*sin(x)
m = 10
Indef_I_m = 1/32*x + 1/64*sin(2*x) + 17/512*sin(3*x) + 1/128*sin(4*x) +
7/2560*sin(5*x) + 1/192*sin(6*x) + 3/1792*sin(7*x) + 1/1152*sin(9*x) +
5/5632*sin(11*x) + 3/6656*sin(13*x) + 1/2560*sin(15*x) +
5/8704*sin(17*x) + 3/9728*sin(19*x) + 1/2688*sin(21*x) +
1/2944*sin(23*x) + 1/6400*sin(25*x) + 1/4608*sin(27*x) +
3/14848*sin(29*x) + 3/15872*sin(31*x) + 5/16896*sin(33*x) +
3/8960*sin(35*x) + 3/9472*sin(37*x) + 1/3328*sin(39*x) +
5/20992*sin(41*x) + 1/5504*sin(43*x) + 1/7680*sin(45*x) +
1/12032*sin(47*x) + 1/12544*sin(49*x) + 1/26112*sin(51*x) +
1/27136*sin(53*x) + 1/28160*sin(55*x) + 13/128*sin(x)
m = 11
Indef_I_m = 51/1024*x + 53/2048*sin(2*x) + 13/768*sin(3*x) + 53/4096*sin(4*x) +
13/1536*sin(6*x) + 1/2048*sin(8*x) + 1/2560*sin(10*x) + 1/3072*sin(12*x)
+ 5/14336*sin(14*x) + 1/4096*sin(16*x) + 5/18432*sin(18*x) +
1/4096*sin(20*x) + 1/5632*sin(22*x) + 5/24576*sin(24*x) +
5/26624*sin(26*x) + 5/28672*sin(28*x) + 1/5120*sin(30*x) +
3/16384*sin(32*x) + 5/34816*sin(34*x) + 1/9216*sin(36*x) +
5/38912*sin(38*x) + 1/10240*sin(40*x) + 1/10752*sin(42*x) +
3/22528*sin(44*x) + 3/23552*sin(46*x) + 1/8192*sin(48*x) +
3/25600*sin(50*x) + 5/53248*sin(52*x) + 1/13824*sin(54*x) +
3/57344*sin(56*x) + 1/29696*sin(58*x) + 1/30720*sin(60*x) +
1/63488*sin(62*x) + 1/65536*sin(64*x) + 1/67584*sin(66*x) +
13/256*sin(x)
so for $m=7$ answer is right compare with [Indef_I_7 via WolframAlpha](http://www.wolframalpha.com/input/?i=integrate+cos%28x%29*cos%282x%29*cos%283x%29*cos%284x%29*cos%285x%29*cos%286x%29*cos%287x%29)
and for $m=8$ answer is incorrect [Indef_I_8 via WolframAlpha](http://www.wolframalpha.com/input/?i=integrate+cos%28x%29*cos%282x%29*cos%283x%29*cos%284x%29*cos%285x%29*cos%286x%29*cos%287x%29*cos%288x%29)
There should be Indef_I_8=$\frac{7x}{128}+\ldots$ and no $\sin(x)$, $\sin(3x)$ in summation, only $\sin(2k)$ for $k=1,2,3,\ldots 18$
Sorry for volumetric calculations !
The question is - Am I right that it is the bug in the symbolic integration?IvanGMon, 17 Jun 2013 06:39:21 -0500https://ask.sagemath.org/question/10248/Integrate with elliptic integral special function in resulthttps://ask.sagemath.org/question/10123/integrate-with-elliptic-integral-special-function-in-result/I'm trying to work with the following integral:
$$\int\sqrt{1-\frac14(\cosh x)^2}\mathrm dx$$
Feeding this to sage as `integrate(sqrt(1-1/4*cosh(x)^2),x)` leaves it pretty much as it stands. [Feeding the same to Wolfram Alpha](http://www.wolframalpha.com/input/?i=integrate%28sqrt%281-1%2F4*cosh%28x%29^2%29%2Cx%29), I get a solution which at least at first glance looks better:
$$\int\sqrt{1-\frac14(\cosh x)^2}\mathrm dx=-\frac12i\sqrt3E\left(ix\Big\vert-\frac13\right)$$
So I wonder:
* **Is there a way to obtain this kind of output using sage?** (This is my main question.)
* In particular, is there a way to manually indicate that a given integral can likely be expressed in terms of elliptic integral functions, and that these would be of interest?
* Are these [elliptic integral functions](http://en.wikipedia.org/wiki/Elliptic_integral) even available at all inside sage? If they are, under what name?
* Is there any benefit in using these special elliptic integral functions, as opposed to (a `numeric_integral` version of) the original integral, in terms of performance or accuracy when dealing with actual numeric data?MvGTue, 14 May 2013 20:43:21 -0500https://ask.sagemath.org/question/10123/Why does this not integratehttps://ask.sagemath.org/question/9571/why-does-this-not-integrate/why is this integral performed:
forget()
var('n')
assume(n>0)
integrate(1/sqrt(1+x^2*n),x,1,2)
i.e. I get:
-arcsinh(sqrt(n))/sqrt(n) + arcsinh(2*sqrt(n))/sqrt(n)
while this is not
forget()
var('n')
assume(n>0)
integrate(1/sqrt(1+x^2/n),x,1,2)
MarkWed, 28 Nov 2012 02:06:22 -0600https://ask.sagemath.org/question/9571/mistake in a indefinite integralhttps://ask.sagemath.org/question/9547/mistake-in-a-indefinite-integral/Hi,
I'm trying to compute the integral:
>integral(log(cot(x)-1),x,0,pi/4)
Sage (with version 5.4 and previous ones) tell that the integral is $-\infty$. But the integral converges, and it is is equal to $\frac{\pi\log(2)}8$.
I think this can be a bug. Maybe it is a Maxima issue, but it should be interesting to find where the bug is, and correct it, if possible.mathematicboyMon, 19 Nov 2012 05:02:30 -0600https://ask.sagemath.org/question/9547/Implementing the basic Fourier-Transformationhttps://ask.sagemath.org/question/9469/implementing-the-basic-fourier-transformation/Hi there!
I'm currently plaing around with sage and I'm really excited about it.
I'd love to do my computations at university and home with a neat opensource-tool instead of the higly prices closed competitors.
Now, the problem I am facing is the symbolif computation of a fourier transformation.
Below is my current naive approach (I'm still learning fourier and complex mathematics, but with large interest!)
x,w,f_0,t = var("x,w,f_0,t")
w = 2*pi*f_0
x(t) = sin(w*t)
integrate(x*exp(-I*w*t),t, -oo, oo)
which results in the following (obviously equal) result:
integrate(e^(-2*I*pi*f_0*t)*sin(2*pi*f_0*t), t, -Infinity, +Infinity)
My expectiation would be an equation without the t (since it has been substituted through integration) and an floating f_0 which I can set according to my desired sine frequency.
Please, could someone tell me, what exactly I am missing here?
Greetings
JakobJakob HolderbaumWed, 24 Oct 2012 23:43:37 -0500https://ask.sagemath.org/question/9469/How do I understand the result of symbolic integralshttps://ask.sagemath.org/question/7574/how-do-i-understand-the-result-of-symbolic-integrals/So now I know how to integrate, but when I type in
sage: deriv=diff((exp(x)-1)/x,x); deriv
e^x/x - (e^x - 1)/x^2
sage: deriv.integrate(x)
-1/x + Ei(x) - gamma(-1, -x)
why don't I get back `(exp(x)-1)/x +C `?
Philipp SchneiderWed, 18 Aug 2010 13:04:12 -0500https://ask.sagemath.org/question/7574/