@@ -102,7 +102,8 @@ We'll need the following imports:
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import matplotlib.pyplot as plt
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plt.rcParams["figure.figsize"] = (11, 5) #set default figure size
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import numpy as np
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- from sympy import *
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+ from sympy import (Symbol, symbols, Eq, nsolve, sqrt, cos, sin, simplify,
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+ init_printing, integrate)
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```
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### An Example
@@ -491,11 +492,38 @@ integrate(cos(ω) * sin(ω), (ω, -π, π))
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We invite the reader to verify analytically and with the `sympy` package the following two equalities:
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$$
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- \int_{-\pi}^{\pi} \cos (\omega)^2 \, d\omega = \frac{\pi}{2}
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+ \int_{-\pi}^{\pi} \cos (\omega)^2 \, d\omega = \pi
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$$
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$$
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- \int_{-\pi}^{\pi} \sin (\omega)^2 \, d\omega = \frac{\pi}{2}
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+ \int_{-\pi}^{\pi} \sin (\omega)^2 \, d\omega = \pi
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$$
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+ ```
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+
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+ ``` {solution-start} complex_ex1
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+ :class: dropdown
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+ ```
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+
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+ Let's import symbolic $\pi$ from ` sympy `
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+
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+ ``` {code-cell} ipython3
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+ # Import symbolic π from sympy
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+ from sympy import pi
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+ ```
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+
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+ ``` {code-cell} ipython3
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+ print('The analytical solution for the integral of cos(ω)**2 \
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+ from -π to π is:')
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+
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+ integrate(cos(ω)**2, (ω, -pi, pi))
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+ ```
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+
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+ ``` {code-cell} ipython3
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+ print('The analytical solution for the integral of sin(ω)**2 \
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+ from -π to π is:')
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+
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+ integrate(sin(ω)**2, (ω, -pi, pi))
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+ ```
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+ ``` {solution-end}
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```
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