On Wednesday, 7 September 2022 at 15:38:44 UTC+1 brombo wrote:
> I calculated an integral g when printed grives:
>
> Piecewise((sin(w/2)/w - 2*cos(w/2)/w**2 + 4*sin(w/2)/w**3, Ne(w, 0)),
> (2/3, True)) + 2*Piecewise((-sin(w/2)/(2*w) + cos(w/2)/w**2 + sin(w/2)/w**3
> - sin(3*w/2)/w**3, Ne(w, 0)), (1/6, True))
>
So this is what we have:
In [13]: w = symbols('w')
In [14]: result = Piecewise((sin(w/2)/w - 2*cos(w/2)/w**2 +
4*sin(w/2)/w**3, Ne(w, 0)), (2/3, True))
...: + 2*Piecewise((-sin(w/2)/(2*w) + cos(w/2)/w**2 + sin(w/2)/w**3 -
sin(3*w/2)/w**3, Ne(w, 0)),
...: (1/6, True))
In [15]: result = nsimplify(result)
In [16]: result
Out[16]:
⎛⎧ ⎛w⎞ ⎛w⎞ ⎛w⎞ ⎞ ⎛⎧ ⎛w⎞ ⎛w⎞ ⎛w⎞
⎛3⋅w⎞ ⎞
⎜⎪sin⎜─⎟ 2⋅cos⎜─⎟ 4⋅sin⎜─⎟ ⎟ ⎜⎪ sin⎜─⎟ cos⎜─⎟ sin⎜─⎟
sin⎜───⎟ ⎟
⎜⎪ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎟ ⎜⎪ ⎝2⎠ ⎝2⎠ ⎝2⎠
⎝ 2 ⎠ ⎟
⎜⎪────── - ──────── + ──────── for w ≠ 0⎟ ⎜⎪- ────── + ────── + ──────
- ──────── for w ≠ 0⎟
⎜⎨ w 2 3 ⎟ + 2⋅⎜⎨ 2⋅w 2 3
3 ⎟
⎜⎪ w w ⎟ ⎜⎪ w w
w ⎟
⎜⎪ ⎟ ⎜⎪
⎟
⎜⎪ 2/3 otherwise⎟ ⎜⎪ 1/6
otherwise⎟
⎝⎩ ⎠ ⎝⎩
⎠
>
> First, how can I extract from g (without cutting and pasting the output)
> what I really want which is:
>
> (sin(w/2)/w - 2*cos(w/2)/w**2 + 4*sin(w/2)/w**3)+2*(-sin(w/2)/(2*w) +
> cos(w/2)/w**2 + sin(w/2)/w**3 - sin(3*w/2)/w**3)
>
In [23]: piecewise_fold(result)
Out[23]:
⎧ ⎛w⎞ ⎛3⋅w⎞
⎪6⋅sin⎜─⎟ 2⋅sin⎜───⎟
⎪ ⎝2⎠ ⎝ 2 ⎠
⎪──────── - ────────── for w ≠ 0
⎨ 3 3
⎪ w w
⎪
⎪ 1 otherwise
⎩
In [24]: piecewise_fold(result).args[0][0]
Out[24]:
⎛w⎞ ⎛3⋅w⎞
6⋅sin⎜─⎟ 2⋅sin⎜───⎟
⎝2⎠ ⎝ 2 ⎠
──────── - ──────────
3 3
w w
>
> Secondly, I can substitute y = w/2. In the resulting expression force
> expansion of sin(3*y). trigsimp doesn't do that. I know the answer is
> supposed to be of the form (sin(y)/y)**3.
>
In [26]: _.subs(w, 2*y)
Out[26]:
3⋅sin(y) sin(3⋅y)
──────── - ────────
3 3
4⋅y 4⋅y
In [27]: _.expand(trig=True)
Out[27]:
3
sin (y)
───────
3
y
--
Oscar
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