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Magazine Problem 3
Consider \(y^{\prime \prime}+\lambda y=0\) subject to \(y^{\prime}(0)=0, y^{\prime}(L)=0\). Show that the eigenfunctions are $$ \left\\{1, \cos \frac{\pi}{L} x, \cos \frac{2 \pi}{L} x, \ldots\right\\} $$ This set, which is orthogonal on \([0, L]\), is the basis for the Fourier cosine series.
Problem 3
In Problems, find the Fourier series of \(f\) on the given interval.
$$
f(x)=\left\\{\begin{array}{lr}
1, & -1
Problem 3
Show that the given functions are orthogonal on the indicated interval. $$ f_{1}(x)=e^{x}, f_{2}(x)=x e^{-x}-e^{-x} ; \quad[0,2] $$
Problem 3
Determine whether the function is even, odd, or neither. $$ f(x)=x^{2}+x $$
Problem 4
Consider \(y^{\prime \prime}+\lambda y=0\) subject to the periodic boundary conditions \(y(-L)=y(L), y^{\prime}(-L)=y^{\prime}(L)\). Show that the eigenfunctions are \(\left\\{1, \cos \frac{\pi}{L} x, \cos \frac{2 \pi}{L} x, \ldots, \sin \frac{\pi}{L} x, \sin \frac{2 \pi}{L} x, \sin \frac{3 \pi}{L} x, \ldots\right\\}\) This set, which is orthogonal on \([-L, L]\), is the basis for the Fourier series.
Problem 4
Determine whether the function is even, odd, or neither. $$ f(x)=x^{3}-4 x $$
Problem 4
In Problems, find the Fourier series of \(f\) on the given interval.
$$
f(x)=\left\\{\begin{array}{lr}
0, & -1
Problem 4
Show that the given functions are orthogonal on the indicated interval. $$ f_{1}(x)=\cos x, f_{2}(x)=\sin ^{2} x ; \quad[0, \pi] $$
Problem 5
Show that the given functions are orthogonal on the indicated interval. $$ f_{1}(x)=x, f_{2}(x)=\cos 2 x ; \quad[-\pi / 2, \pi / 2] $$
Problem 5
Find the complex Fourier series of \(f\) on the given interval.
$$
f(x)=x, 0
