Function Plot

from smpl import plot
import numpy as np
import smpl
import math
smpl.__version__

without uncertainties

\(\dot x = 1- \exp(- x^2)\)

Fixed point \(x = 0\) and

\(\ddot x = -2x \exp(-x^2) \implies \ddot x(x = 0)=0\)

only metastable for \(x\lt0\)

plot.function( lambda x : 1- np.exp(-x**2), xaxis="$x$", yaxis="$\\dot x$",xmin=-10, xmax=10 )

\(\dot x = \ln x\)

Fixed point \(x = 1\)

\[\ddot x = \frac{1}{x} \implies \ddot x(x=1) = 1 > 0\]

\(\implies\) unstable

plot.function( lambda x : np.log(x), xaxis="$x$", yaxis="$\\dot x$",xmin=0.1, xmax=5 )

\(\dot x = -\tan x\)

Fixed points for \(x=0\) or \(x=\pm n\pi\) with \(n\in \mathbb{N}\)

\[\ddot x = -\frac{1}{\cos^2(x)}\]

\[\ddot x(x=0) = -1 \lt 0\]

\[\ddot x(x=n \pi) = -1 \lt 0\]

\(\implies\) stable

plot.function( lambda x : -np.tan(x), xaxis="$x$", yaxis="$\\dot x$",xmin=0.1, xmax=5,steps=100 )

with uncertainties

import uncertainties as unc
a = unc.ufloat(1,0.1)

plot.function(lambda x : 1- a*np.exp(-x**2), xaxis="$x$", yaxis="$\\dot x$",xmin=-1, xmax=1,sigmas=1 )

Complex

from smpl.stat import fft
y = np.sin(np.arange(256))
print(len(fft(y)))
plot.data(*fft(y),label="FFT",fmt="-")

from smpl.stat import fft
plot.data(*fft(np.sin(np.arange(256))),*fft(np.sin(1/np.pi*np.arange(100))),label="FFT",fmt="-")

without xmin and xmax

xmin and xmax will have to be guessed

from smpl import plot
plot.function(lambda x: x**2,)

from smpl import plot
import numpy as np
def f(x):
    return np.exp(x)
plot.function(f,label="exp")

from smpl import plot
from smpl import functions as f
def gauss(x):
    """Gauss"""
    return f.gauss(x,0,1,3,0)
plot.function(gauss)

def gauss(x):
    return np.arctan(x)
plot.function(gauss)

def gauss(x):
    return np.tan(x)
plot.function(gauss)

def gauss(x):
    return np.log(x)
plot.function(gauss)

def gauss(x):
    return x**3+5*x**2-2
plot.function(gauss)

def gauss(x):
    return x**0.5
plot.function(gauss)

Guessing the interesting regions of a function can’t always be correct/satisfactory, especially in numerical unstable regions:

c=299792458#m/s
h=4.13566769692*10**-15#eVs
kb=8.617333262*10**-5#eV/K
T=273
def Strahlungsgesetz(x):
    return 8*np.pi/c**3*h*x**3/(np.exp((h*x)/(kb*T))-1)
plot.function(Strahlungsgesetz,xaxis="$x$", yaxis="$\\dot x$")

plot.function(Strahlungsgesetz,xaxis="$x$", yaxis="$\\dot x$",xmin=1e-7-2e-2,xmax=1e-7+2e-2)

plot.function(Strahlungsgesetz,xaxis="$x$", yaxis="$\\dot x$",xmin=1,xmax=0.3e15)