Department of Mathematics University of Utah\ 1.5 ] Written Qualifying Examination in Applied Mathematics

Construct the Green function for the equation \[ -u''+u=f(x)\qquad (-\infty \le x\le +\infty ),\qquad u(x)\to 0\text { as }x\to \pm \infty . \]

Solve the differential equation in the sense of distributions \[ xu'=\delta (x^4-1),\qquad x\in [-2,2],\qquad u(-2)=u(2)=0. \] Discuss why two boundary conditions are provided for the first order differential equation.

Suppose that \(T\) is a transformation of a complete metric space \(X\), and \(T^n\) (for a certain positive integer \(n\)) is a contraction with fixed point \(a\). Show that

1. \(a\) is a fixed point of \(T\);
2. \(T\) has only one fixed point;
3. the sequence of successive iterations \(u_0\), \(u_1=Tu_0\), \(u_2=Tu_1,\ldots \) tends to \(a\) for any initial element \(u_0\in X\).

Let \(C\) be a closed convex set in a Hilbert space \(H\), and let \(u\) be an arbitrary vector in \(H\). Show that there exists one and only one element in \(C\) which is closest to \(u\).

Prove the following statement. In an arbitrary real Hilbert space \(H\) an orthonormal system \((\phi _1,\phi _2,\ldots ,\phi _n,\ldots )\) is complete if and only if the equality \[ \lVert f\rVert ^2=\sum _{n=1}^{\infty }\langle f,\phi _n\rangle ^2 \] holds for any \(f\in H\).

Evaluate the integral \[ I=\int _0^\infty \frac {\sin x}{x}\,dx. \]

Evaluate the integral \[ I=\int _0^\infty \frac {x^\alpha }{x^2+1}\,dx \] where \(\alpha \) is a parameter, \(-1<\alpha <1\).

Find a conformal transformation of the domain \(D\) between the unit circle \(\Gamma _1=\{z=x+iy: |z|=1\}\) and the vertical line \(\Gamma _2=\{z=x+iy: x=5/3\}\) onto a domain between two concentric circles.

Liouville’s theorem states that a function analytic and bounded in the complex plane is necessarily a constant.

Prove this theorem. Using it prove the Fundamental theorem of algebra: Every polynomial \(P(z)\) which is not a constant has at least one complex root.

Find the bounded solution \(u(x,y)\) of the Laplace equation in the sector \[ D=\{(r,\theta ): 0<\theta <3\pi /4\}\qquad (x+iy=z=re^{i\theta }) \] satisfying the boundary conditions \[ u=0\text { when }\theta =0,\qquad u=5\text { when }\theta =\frac {3\pi }{4}. \]