In [1]:

%pylab inline

Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].
For more information, type 'help(pylab)'.

In [2]:

from sympy import Symbol, fresnels, fresnelc, oo, I, re, im, series, Rational, sin, cos, exp, plot
from sympy.plotting import plot, plot_parametric
from matplotlib.pyplot import figsize

Plot of the two Fresnel integrals $S(x)$ and $C(x)$

In [3]:

x = Symbol("x")

In [4]:

plot(fresnels(x), fresnelc(x), (x, 0, 8))

Out[4]:

<sympy.plotting.plot.Plot at 0xa3be1ac>

The Cornu spiral defined as the parametric curve $u(t),v(t) := C(t), S(t)$

In [5]:

figsize(8,8)
plot_parametric(fresnelc(x), fresnels(x))

Out[5]:

<sympy.plotting.plot.Plot at 0xb22c76c>

Compute and plot the leading order behaviour around $x=0$

In [6]:

ltc = series(fresnelc(x), x, n=2).removeO()
lts = series(fresnels(x), x, n=4).removeO()

In [7]:

lts, ltc

Out[7]:

(pi*x**3/6, x)

In [8]:

figsize(4,4)
plot(fresnels(x), lts, (x, 0, 1))
plot(fresnelc(x), ltc, (x, 0, 1))

Out[8]:

<sympy.plotting.plot.Plot at 0x101b4bec>

Compute and plot the asymptotic series expansion at $x=\infty$

In [9]:

# Series expansion at infinity is not implemented yet
#ats = series(fresnels(x), x, oo)
#atc = series(fresnelc(x), x, oo)

In [10]:

# However we can use the well known values
ats = Rational(1,2) - cos(pi*x**2/2)/(pi*x)
atc = Rational(1,2) + sin(pi*x**2/2)/(pi*x)

In [11]:

figsize(4,4)
plot(fresnels(x), ats, (x, 6, 8))
plot(fresnelc(x), atc, (x, 6, 8))

Out[11]:

<sympy.plotting.plot.Plot at 0xda47a8c>

In [12]:

alpha = Symbol("alpha")
r = 3.0
circ = r*exp(1.0j*alpha)

In [13]:

figsize(8,8)
plot_parametric(re(fresnelc(circ)), im(fresnelc(circ)), (alpha, 0, 2*pi))

Out[13]:

<sympy.plotting.plot.Plot at 0xb15a4ec>

In [13]: