avery_tutorial_sage_2
#Create some symbolic variables to be used here
#Note that x is the only predefined symbolic variable.
#Turn on formatted output by checking the Typeset box above
var("a b c d y z")
#Create a function using basic notation.
#Notice the difference between print output and typeset output
f(x) = x^2 * sin(x)^2
print 1/x, f(x)
1/x, f(x)
1/x x^2*sin(x)^2
#Remember how to evaluate exact vs floating point expressions
f(2); f(2.)
#You can differentiate and integrate functions just like ordinary
expressions
#Note that diff can handle multiple derivatives in sequence
D1 = diff(f(x), x, 2)
I1 = integrate(f(x), x)
I2 = integrate(f(x), x, 0, 2*pi).factor()
E1 = (x^3 * y^4).diff(x, 1, y, 2)
E2 = (x^3 * y^4).diff(y, 2, x, 1)
D1; I1; I2; E1; E2
#x first, then y
#y first, then x
#Power series using the taylor() function. Let's do a simple one first.
#Expand exp(x) around 0 to 5th order
taylor(exp(x), x, 0, 5)
#Here's a more complicated one expanded to 10th order!
taylor(exp(tan(x)), x, 0, 10)
#Expand our previously defined function f(x) = x^2 * sin(x)^2
taylor(f(x), x, 0, 12).factor()
#Let's expand the potential function phi(x,t) = 1 / sqrt(1 - 2*t*x + x^2)
#The coefficients of x^n are n^th order Legendre polynomials in t.
#That's why phi(x,t) is called a "generating function" for Legendre
polynomials.
var("t")
phi(x,t) = 1 / sqrt(1 - 2*t*x + x^2)
taylor(phi(x,t), x, 0, 5)
#Indefinite or definite integral of an expression or function
integrate(sin(x)^2, x); integrate(sin(x)^2, x, 0, pi)
#Sage represents +infinity as oo.
oo; -oo; tan(pi/2); atan(oo)
#Integrate exp(-x^2) from -infinity to +infinity
integrate(exp(-x^2), x, -oo, oo)
#Now do some sums over powers of integers
var("n N")
sum(n, n, 1, N).factor(); sum(n^2, n, 1, N).factor(); sum(n^3, n, 1,
N).factor()
#How about some sums over large powers of integers
sum(n^8, n, 1, N).factor(); sum(n^10, n, 1, N).factor()
#Do some infinite sums over negative powers of integers.
#This defines the Riemann zeta function: zeta(k) = sum(1/n^k, n, 1, oo)
#These sums occur when integrating photon energy and number distributions
sum(1/n^2, n, 1, oo); zeta(2); sum(1/n^3, n, 1, oo); zeta(3); sum(1/n^4,
n, 1, oo); zeta(4);
#Sum over powers of odd integers and powers of alternating sign integers
sum(1/(2*n-1)^2, n, 1, oo); sum( (-1)^(n-1)/n^2, n, 1, oo )
#Summing an infinite series
assume(abs(x) < 1);
sum(x^n, n, 0, oo); sum(x^(2*n+1), n, 0, oo); sum( x^n/factorial(n), n,
0, oo)
#Try evaluating sin(k*pi), cos(k*pi) without knowing that k is an integer
var("k")
sin(k*pi), cos(k*pi)
#Now tell Sage that k is a positive integer. Unfortunately, it doesn't
give more info.
assume(k, "integer"); assume(k > 0)
sin(k*pi), cos(k*pi)
#Try integrating sin(m*x) without knowing that m is an integer
var("m")
integrate(sin(m*x), x, 0, 2*pi).factor()
#Now tell Sage that m is an integer and integrate again
assume(m, "integer")
integrate(sin(m*x), x, 0, 2*pi)
#We can find all assumptions using the assumptions() command
assumptions()
#Or just find out about a particular variable
assumptions(k)
#You can delete particular assumptions using the forget() function
(k > 0).forget(); assumptions(k)
#You can also forget all assumptions.
forget(); assumptions()
#Create variables a and n and tell sage they are positive
var("a n"); assume(a > 0); assume(n > 0); assume(n, "integer")
#Now we can integrate with them. Note that we forced n to be integer, but
the gamma
#function does not actually require that. This is a shortcoming of sage.
f1 = exp(-a*x)
f2 = 1 / (x^2 + a^2)
f3 = x^n * exp(-a*x)
integrate(f1, x, 0, oo); integrate(f2, x, -oo, oo); integrate(f3, x, 0,
oo)
#You can list all variables associated with function or expression with
the variables() function
h(x) = x^2 * cos(x)
g(a,b) = a^2 + b^2
h().variables(); g.variables()
#Sage supports a large number of special functions, including orthogonal
polynomials.
#Look at standard Legendre polynomials order 0 - 5.
#The factor() suffix puts the polynomials in recognizable form
P0 = legendre_P(0,x)
P1 = legendre_P(1,x)
P2 = legendre_P(2,x).factor()
P3 = legendre_P(3,x).factor()
P4 = legendre_P(4,x).factor()
P5 = legendre_P(5,x).factor()
P0; P1; P2; P3; P4; P5
#Hermite polynomials are used in the solution of Schrodinger's equation
for the harmonic oscillator
H0 = hermite(0,
H1 = hermite(1,
H2 = hermite(2,
H3 = hermite(3,
H4 = hermite(4,
H5 = hermite(5,
H0; H1; H2; H3;
x)
x)
x)
x)
x)
x)
H4; H5
#Laguerre polynomials are used in the solution of Schrodinger's radial
equation for hydrogen
L0 = laguerre(0,x)
L1 = laguerre(1,x)
L2 = laguerre(2,x)
L3 = laguerre(3,x)
L4 = laguerre(4,x)
L5 = laguerre(5,x)
L0; L1; L2; L3; L4; L5