#12628: Integration of Product of Sin(nz)/(nz) says divergent, but is not
divergent
------------------------+---------------------------------------------------
Reporter: dkrenn | Owner: burcin
Type: defect | Status: new
Priority: major | Milestone: sage-5.0
Component: calculus | Keywords: integration, divergent, infinity
Work_issues: | Upstream: N/A
Reviewer: | Author:
Merged: | Dependencies:
------------------------+---------------------------------------------------
Description changed by dkrenn:
Old description:
> We have
> {{{
> sage: integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7),z,0,oo)
> Traceback (most recent call last)
> ...
> ValueError: Integral is divergent.
> }}}
> but the value of that integral is {{{22/315*pi}}}, see, for example,
> evaluations in Mathematica below.
>
> Splitting the integral gives
> {{{
> sage: integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7),z,0,1)
> 1/105*integrate(sin(3*z)*sin(5*z)*sin(7*z)*sin(z)/z^4, z, 0, 1)
> sage: integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7),z,1,oo)
> 256/105*I*gamma(-3, -16*I) - 49/30*I*gamma(-3, -14*I) - 25/42*I*gamma(-3,
> -10*I) + 32/105*I*gamma(-3, -8*I) - 9/70*I*gamma(-3, -6*I) +
> 4/105*I*gamma(-3, -4*I) - 1/210*I*gamma(-3, -2*I) + 1/210*I*gamma(-3,
> 2*I) - 4/105*I*gamma(-3, 4*I) + 9/70*I*gamma(-3, 6*I) -
> 32/105*I*gamma(-3, 8*I) + 25/42*I*gamma(-3, 10*I) + 49/30*I*gamma(-3,
> 14*I) - 256/105*I*gamma(-3, 16*I) + 1/2520
> }}}
> so we see that the integral from 0 to 1 is not evaluated.
>
> Here are some more examples and the comparison to Mathematica:
> {{{
> sage: var('z,n')
> (z, n)
> sage: f(z,n) = sin(n*z)/(n*z)
> sage: integrate(f(z,1),z,0,oo)
> 1/2*pi
> sage: integrate(f(z,1)*f(z,3),z,0,oo)
> 1/6*pi
> sage: integrate(f(z,1)*f(z,3)*f(z,5),z,0,oo)
> 1/10*pi
> sage: integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7),z,0,oo)
> Traceback (most recent call last)
> ...
> ValueError: Integral is divergent.
> sage: integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7)*f(z,9),z,0,oo)
> 3677/72576*pi
> sage: integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7)*f(z,9)*f(z,11),z,0,oo)
> Traceback (most recent call last)
> ...
> ValueError: Integral is divergent.
> sage:
> integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7)*f(z,9)*f(z,11)*f(z,13),z,0,oo)
> 193359161/6227020800*pi
> sage:
> integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7)*f(z,9)*f(z,11)*f(z,13)*f(z,15),z,0,oo)
> Traceback (most recent call last)
> ...
> ValueError: Integral is divergent.
> }}}
>
> {{{
> In[1]:= f[z_,n_]:=Sin[n z]/(n z)
>
> In[2]:= Integrate[f[z,1],{z,0,Infinity}]
>
> Pi
> Out[2]= --
> 2
>
> In[3]:= Integrate[f[z,1]*f[z,3],{z,0,Infinity}]
>
> Pi
> Out[3]= --
> 6
>
> In[4]:= Integrate[f[z,1]*f[z,3]*f[z,5],{z,0,Infinity}]
>
> Pi
> Out[4]= --
> 10
>
> In[5]:= Integrate[f[z,1]*f[z,3]*f[z,5]*f[z,7],{z,0,Infinity}]
>
> 22 Pi
> Out[5]= -----
> 315
>
> In[6]:= Integrate[f[z,1]*f[z,3]*f[z,5]*f[z,7]*f[z,9],{z,0,Infinity}]
>
> 3677 Pi
> Out[6]= -------
> 72576
>
> In[7]:=
> Integrate[f[z,1]*f[z,3]*f[z,5]*f[z,7]*f[z,9]*f[z,11],{z,0,Infinity}]
>
> 48481 Pi
> Out[7]= --------
> 1247400
>
> In[8]:=
> Integrate[f[z,1]*f[z,3]*f[z,5]*f[z,7]*f[z,9]*f[z,11]*f[z,13],{z,0,Infinity}]
>
> 193359161 Pi
> Out[8]= ------------
> 6227020800
>
> In[9]:=
> Integrate[f[z,1]*f[z,3]*f[z,5]*f[z,7]*f[z,9]*f[z,11]*f[z,13]*f[z,15],{z,0,Infinity}]
>
> 5799919 Pi
> Out[9]= ----------
> 227026800
> }}}
New description:
We have
{{{
sage: var('z,n')
(z, n)
sage: f(z,n) = sin(n*z)/(n*z)
sage: integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7),z,0,oo)
Traceback (most recent call last)
...
ValueError: Integral is divergent.
}}}
but the value of that integral is {{{22/315*pi}}}, see, for example,
evaluations in Mathematica below.
Splitting the integral gives
{{{
sage: integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7),z,0,1)
1/105*integrate(sin(3*z)*sin(5*z)*sin(7*z)*sin(z)/z^4, z, 0, 1)
sage: integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7),z,1,oo)
256/105*I*gamma(-3, -16*I) - 49/30*I*gamma(-3, -14*I) - 25/42*I*gamma(-3,
-10*I) + 32/105*I*gamma(-3, -8*I) - 9/70*I*gamma(-3, -6*I) +
4/105*I*gamma(-3, -4*I) - 1/210*I*gamma(-3, -2*I) + 1/210*I*gamma(-3, 2*I)
- 4/105*I*gamma(-3, 4*I) + 9/70*I*gamma(-3, 6*I) - 32/105*I*gamma(-3, 8*I)
+ 25/42*I*gamma(-3, 10*I) + 49/30*I*gamma(-3, 14*I) - 256/105*I*gamma(-3,
16*I) + 1/2520
}}}
so we see that the integral from 0 to 1 is not evaluated.
Here are some more examples and the comparison to Mathematica:
{{{
sage: var('z,n')
(z, n)
sage: f(z,n) = sin(n*z)/(n*z)
sage: integrate(f(z,1),z,0,oo)
1/2*pi
sage: integrate(f(z,1)*f(z,3),z,0,oo)
1/6*pi
sage: integrate(f(z,1)*f(z,3)*f(z,5),z,0,oo)
1/10*pi
sage: integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7),z,0,oo)
Traceback (most recent call last)
...
ValueError: Integral is divergent.
sage: integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7)*f(z,9),z,0,oo)
3677/72576*pi
sage: integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7)*f(z,9)*f(z,11),z,0,oo)
Traceback (most recent call last)
...
ValueError: Integral is divergent.
sage: integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7)*f(z,9)*f(z,11)*f(z,13),z,0,oo)
193359161/6227020800*pi
sage:
integrate(f(z,1)*f(z,3)*f(z,5)*f(z,7)*f(z,9)*f(z,11)*f(z,13)*f(z,15),z,0,oo)
Traceback (most recent call last)
...
ValueError: Integral is divergent.
}}}
{{{
In[1]:= f[z_,n_]:=Sin[n z]/(n z)
In[2]:= Integrate[f[z,1],{z,0,Infinity}]
Pi
Out[2]= --
2
In[3]:= Integrate[f[z,1]*f[z,3],{z,0,Infinity}]
Pi
Out[3]= --
6
In[4]:= Integrate[f[z,1]*f[z,3]*f[z,5],{z,0,Infinity}]
Pi
Out[4]= --
10
In[5]:= Integrate[f[z,1]*f[z,3]*f[z,5]*f[z,7],{z,0,Infinity}]
22 Pi
Out[5]= -----
315
In[6]:= Integrate[f[z,1]*f[z,3]*f[z,5]*f[z,7]*f[z,9],{z,0,Infinity}]
3677 Pi
Out[6]= -------
72576
In[7]:=
Integrate[f[z,1]*f[z,3]*f[z,5]*f[z,7]*f[z,9]*f[z,11],{z,0,Infinity}]
48481 Pi
Out[7]= --------
1247400
In[8]:=
Integrate[f[z,1]*f[z,3]*f[z,5]*f[z,7]*f[z,9]*f[z,11]*f[z,13],{z,0,Infinity}]
193359161 Pi
Out[8]= ------------
6227020800
In[9]:=
Integrate[f[z,1]*f[z,3]*f[z,5]*f[z,7]*f[z,9]*f[z,11]*f[z,13]*f[z,15],{z,0,Infinity}]
5799919 Pi
Out[9]= ----------
227026800
}}}
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12628#comment:1>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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