%pylab inline

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

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 $C(x)$

x = Symbol("x")

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

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

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

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

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

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

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

lts, ltc

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

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

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

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

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

# 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)

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

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

Another nice example of a parametric plot

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

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

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