An assorted selection of example calculations using Yacas

1. Show that Integrate(-Pi,Pi) (Sin(n*x)*Cos(m*x)) is Pi*Delta(n,m) : 
In> Simplify(Integrate(x,-Pi,Pi)Sin(x)*Sin(2*x)) 
 Out> 0 
In> Simplify(Integrate(x,-Pi,Pi)Sin(2*x)*Sin(2*x)) 
 Out> Pi 
In>  Simplify(Integrate(x,-Pi,Pi)Sin(5*x)*Sin(5*x)) 
 Out> Pi 
In>  Simplify(Integrate(x,-Pi,Pi)Cos(x)*Cos(2*x)) 
 Out> 0 
In> Simplify(Integrate(x,-Pi,Pi)Cos(2*x)*Cos(2*x)) 
 Out> Pi 
In>  Simplify(Integrate(x,-Pi,Pi)Cos(5*x)*Cos(5*x)) 
 Out> Pi 
In>  Simplify(Integrate(x,-Pi,Pi)Sin(x)*Cos(2*x)) 
 Out> 0 
In> Simplify(Integrate(x,-Pi,Pi)Sin(2*x)*Cos(2*x)) 
 Out> 0 
In> Simplify(Integrate(x,-Pi,Pi)Sin(5*x)*Cos(5*x)) 
 Out> 0 

2. Get the first 5 coefficients of the Fourier series of x^2
on the domain -Pi to Pi. This should be (1/Pi)*Sum(n,0,4)a_n * Cos(n*x) : 
In> Fourier(_n,_f)<--1/Pi*Integrate(x,-Pi,Pi)f*Cos(n*x) 
 Out> True 
In>  TableForm(Simplify(Table(Fourier(n,x^2),n,0,5,1))) 
 (2*Pi^2)/3
-4
1
-4/9
1/4
-4/25
Out> True 

3. Check that f:=x*Exp(-x/2) is a solution to the equation H(f)=E f
where E is a constant and H is D(x)D(x)f + f/x : 
In>  H(f):=Deriv(x)Deriv(x)f+f/x 
In>  f:=x*Exp(-x/2) 
In> res:=H(f) 
In> PrettyForm(Simplify(res)) 

       /  / x \ \
x * Exp| -| - | |
       \  \ 2 / /
-----------------
        4        

In> PrettyForm(Simplify(res/f)) 

1
-
4


4. Show that the first few terms of the Taylor series expansion
of Sin(x) and Cos(x-Pi/2) are the same : 
In> ans1:=Taylor(x,0,8)Sin(x) 
In> PrettyForm(ans1) 

     3    5      7 
    x    x      x  
x - -- + --- - ----
    6    120   5040

In> ans2:=Taylor(x,0,8)Cos(x-Pi/2) 
In> PrettyForm(ans2) 

     3    5      7 
    x    x      x  
x - -- + --- - ----
    6    120   5040

In> ans1-ans2 
 Out> 0 

5. Determine a polynomial that goes through the points
(x,y) = { (-2,4), (1,1), (3,9) } and show that it is in fact x^2 : 
In> ans:=LagrangeInterpolant({-2,1,3},{4,1,9},x) 
In> PrettyForm(ans) 

4 * ( x - 1 ) * ( x - 3 )   ( x - -2 ) * ( x - 3 )   
------------------------- - ---------------------- + 
           15                         6              

9 * ( x - -2 ) * ( x - 1 )
--------------------------
            10            

In> PrettyForm(Simplify(ans)) 

 2
x 

5  examples shown