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