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798 APPENDIX A A MAPLE Primer

This is the matrix of Example 9.3. Now
eigenvects(A);

gives the output
[1,2,[1,0,0]],[−1,1,[1,2,−4]]
This gives eigenvalue 1 with multiplicity 2, and every eigenvector associated with 1 is a
multiple of (1,0,0). Eigenvalue −1 has multiplicity 1 and eigenvector (1,2,−4).
Now let B be the 3 × 3 matrix of Example 9.5:
B:=Matrix(3,3,[[5,-4,4],[12,-11,12],[4,-4,5]]);

The command
eigenvals(B);
gives the list 1,1,−3 of eigenvalues, 1 having multiplicity 2, and −3 having multiplicity 1.
Next use
eigenvects(B);
to obtain the output

[1,2,[1,1,0],[−1,0,1]],[−3,1,[1,3,1]]
giving two linearly eigenvectors associated with eigenvalue −3.

A.5 Integral Transforms

MAPLE has subroutines for several integral transforms. To load these, enter

with(inttrans);
To take the Laplace transform of f (t),use
laplace(f(t),t,s);
in which f (t) may have been loaded previously, or can be specified in the command. For
example,
laplace(t cos(t),t,s);
∗
2
2
2
returns the transform (s − 1)((s + 1) ) of t cos(t).
2
2
For the inverse Laplace transform of 1/((s + 4) ),use
invlaplace(1/((s∧2+4)∧2),s,t);
to obtain (1/16)sin(2t) − (1/8)cos(2t).
For the Fourier, Fourier sine and Fourier cosine transforms, the commands are similar,
except replace laplace with fourier, fouriersin,or fouriercos. For example, for
the Fourier transform of e −|t| ,use
fourier(exp(-abs(t)),t,w);
2
to obtain the transform 2/(1 + w ). To compute the inverse Fourier transform of 1/(1 + w),use
invfourier((1/(1 + w),w,t);
1
−it
to obtain the inverse transform ie (2H(t) − 1), where H(t) is the Heaviside function.
2

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October 14, 2010 15:43 THM/NEIL Page-798 27410_24_appA_p789-800