A new world
This is my first entry in a new blog. I'm trying to create this blog using Pelican, a tool for automatically creating a website for the blog. Articles on the internet aren't all that helpful in describing exactly how to do this. The cost is $\$5.30$.
I'm sure, however, that I will eventually figure it all out.
Here is an in-line equation $\sqrt{3x - 1} + (1 + x^2)^3$ in the body of the text. Here is a displayed equation.
$$x = \frac{-b \pm\sqrt{b^2 - 4ac}}{2a}$$
where $a, b$ and $c$ are constants. The formula solves the equation $ax^2 + bx + c = 0$. One thing we use this formula for is solving quadratic equations. Here are some aligned equations:
$$\begin{align} 3x^2 + 4x - 1 &= 0\\ 2x + 3xy - y &= 0\\ y^2 - 2x^2 + 9 &= 0 \end{align}$$
To solve the formula $ax^2 + bx + c = 0$, use the quadratic formula. We use the quadratic equation to solve it!
![[2in][2in][](http://jakevdp.github.io/images/galaxy.jpg "[2in][2in][")
An Ipython Notebook
import scipy as sp
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
%matplotlib inline
from __future__ import division
from scipy.integrate import quad
def f(x):
return sp.sin(x)
def g(f,a):
pi = sp.pi
return 1/pi * quad(lambda x: f(x)*sp.cos(x),-a,a)[0]
g(f,sp.pi)
def a(f, k):
return (1/sp.pi) * quad(lambda x: f(x)*sp.cos(k*x),-sp.pi,sp.pi)[0]
def b(f, k):
return (1/sp.pi) * quad(lambda x: f(x)*sp.sin(k*x),-sp.pi,sp.pi)[0]
def f(x):
return sp.exp(x)
def F(x,n):
g = lambda x: a(f,0)/2 + sum([a(f,k)*sp.cos(k*x) for k in range(1,n)] + [b(f,k)*sp.sin(k*x) for k in range(1,n+1)])
return g
xrng = sp.linspace(-sp.pi, sp.pi,200)
plt.plot(xrng, F(f,40)(xrng),lw=2)
plt.plot(xrng,sp.exp(xrng),lw=8,alpha=0.5)
def fourier(f,N,x):
coeffs = np.array([quad(f(x)*np.exp(1j*n*x),-np.pi,np.pi) for n in N])
return coeffs
def g(x):
return np.exp(x)
def myint(f):
return quad(lambda x: f(x),0,2)[0]
def g(x):
return 5*x
print(myint(g))
xrng = sp.linspace(-sp.pi,sp.pi,200)
plt.plot(xrng,fourier(lambda x: x**3,20)(xrng),lw=3)
def func(x):
return np.where(np.abs(x) >= 1, 0., np.abs(x) - 1.0)
def cn(x, y, n, period):
c = y * np.exp(-1j * 2. * np.pi * n * x / period)
return c.sum()/c.size
def f(x, y, Nh, period):
rng = np.arange(0.5, Nh+0.5)
coeffs = np.array([cn(x,y,i,period) for i in rng])
f = np.array([2. * coeffs[i] * np.exp(1j*2*i*np.pi*x/period) for i in rng])
return f.sum(axis=0)
a, b = 1, 4
N = 100
Nh = 10
def func(x):
return x*sp.log(x)
time = np.linspace( a, b, N )
y = func(time)
period = np.abs(a-b)
y2 = f(time,y,Nh,period).real
fig = plt.figure()
ax = fig.add_subplot(1,1,1)
ax.plot(time, y, lw=3)
ax.plot(time, y2, lw=3)
plt.show()
a = np.array([(x+1j)*(1-x*1j) for x in range(0,10)],dtype='complex')
print(a)
print(a.real)
print(a.imag)
print(a.sum()/a.max())
print(a.min())
print(a.max())
print(a.sum(), a.sum()/a.size)
rng = np.arange(.5, 8+.5)
print(rng)
print(a.size)
Nh = 10
rng = np.arange(0.5, Nh+0.5)
print(rng)
coeffs = np.array([i for i in rng])
print(coeffs)
f = np.array([(i,coeffs[i]) for i in rng])
print(coeffs[0],coeffs[0.5],coeffs[2],coeffs[2.1],coeffs[2.2],coeffs[2.3],coeffs[2.8],coeffs[2.99],coeffs[3.001])