```python
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt
%matplotlib widget
```

# Fourier Transforms
    - The discrete Fourier transform is defined as
    
$$ y[k] = \sum_{n=0}^{N-1} e^{-2{\pi}in(k/N)}\,x[n]$$

    - k / N represents a specific frequency (dimensionless)
    - can be converted to a frequency(Hz) if you know the spacing in x


```python
x = np.linspace(-2*np.pi, 2*np.pi, 100)
y = np.cos(x) + 1j*np.sin(x)
eix = np.exp(1j*x)
eix_ = np.exp(-1j*x)
#plt.plot(x,y)
#plt.plot(x,eix+1,label=r'$e^i$')
plt.plot(x,np.sin(x),label=r'$\sin(x)$')
plt.plot(x,np.sin(3*x),label=r'$\sin(3x)$')
plt.plot(x,np.sin(-3*x),label=r'$\sin(-3x)$')
#plt.plot(x,np.exp(-1j*x),label=r'$e^-i$')
plt.legend()
```




    <matplotlib.legend.Legend at 0x7feaace8bd90>





<div style="display: inline-block;">
    <div class="jupyter-widgets widget-label" style="text-align: center;">
        Figure
    </div>
    
</div>




```python
import sympy as sp
N = sp.symbols('N')
e = (1+1/N)**N
(e**2).apart()
```




$\displaystyle \left(1 + \frac{1}{N}\right)^{2 N}$




```python

```


```python

```

fd2ecd6f9e829a75253397e9ab0f52fad0e4751f
ipynb
Jupyter Notebook
python/matplotlib/vector/basic/scipy-intergrate.ipynb
karng87/nasm_game
a97fdb09459efffc561d2122058c348c93f1dc87
MIT
python/matplotlib/vector/basic/scipy-intergrate.ipynb
karng87/nasm_game
a97fdb09459efffc561d2122058c348c93f1dc87
MIT
python/matplotlib/vector/basic/scipy-intergrate.ipynb
karng87/nasm_game
a97fdb09459efffc561d2122058c348c93f1dc87
MIT
Qwen/Qwen-72B
1. YES
2. YES