UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Applied Mathematics January 3, 2019

Let \((e_n)\) be an orthonormal sequence of a Hilbert space \(H\). Show that \(e_n\to 0\) weakly.

Let \(T:X\to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that the spectrum \(\sigma (T)\) lies in the complex plane disk: \[ \{\lambda \in \mathbb {C}:\lvert \lambda \rvert \le \lVert T\rVert \}. \]

Let \(T\) be a bounded linear operator on a complex Hilbert space \(H\). Show that the operator \(I+T^*T\) is injective.

Let \(H\) be a Hilbert space. Recall that an operator \(T\in B(H)\) is Hilbert-Schmidt if there exits a complete orthonormal sequence, \((e_n)\in H\) such that \[ \sum _{n=1}^{\infty }\lVert Te_n\rVert ^2<\infty . \] Show that every Hilbert-Schmidt operator is compact.

Consider the distribution \(u\in \mathcal {D}'(\mathbb {R})\) defined by \[ \langle u,\phi \rangle =\int _{-\infty }^{\infty }\lvert x\rvert \phi (x)\,dx, \qquad \phi \in \mathcal {D}(\mathbb {R}). \] Find its derivative, \(\partial u\).

Prove the fundamental theorem of algebra:

Every polynomial of degree \(n(\ge 1)\) has exactly \(n\) zeros (each zero being counted according to its multiplicity).

The following function \(f(x)\) [\(x\) is a real variable] is expanded in a series in powers of \((x+3)\). Find the radius of convergence of the series in each case. \[ \begin {array}{rcl} \text {(a)}\quad f(x)&=&e^{x^2},\\[1.4ex] \text {(b)}\quad f(x)&=&\left (\dfrac {\sin x-3}{\sin x-2}\right )^2,\\[1.8ex] \text {(c)}\quad f(x)&=&\dfrac {\sin (x+1)}{(x+1)(x+3)}. \end {array} \]

Evaluate the integrals \[ \begin {array}{rcll} \text {(a)}&\displaystyle \int _{-\infty }^{\infty }\frac {\sin ax}{x}\,dx &&\text {(\(a\) is a real parameter)},\\[2.2ex] \text {(b)}&\displaystyle \int _0^\infty \frac {x^\alpha }{x+1}\,dx &&\text {(\(\alpha \) is a real parameter, \(-1<\alpha <0\))}. \end {array} \]

1. Consider a 2-dimensional map \((x,y)\to (u,v)\).

   Explain: If the map is given by an analytic function [\(u+iv=f(x+iy)\)] and at some point \(z_0=x_0+iy_0\) the derivative \(f'(z_0)\ne 0\), then this map preserves small shapes in the vicinity of \(z_0\).

2. Does there exist a conformal transformation \(u+iv=f(x+iy)\) that maps triangle \[ \Delta =\{(x,y):0<x<1,\ 0<y<1,\ x+y<1\} \] to circle \[ D=\{(u,v):u^2+v^2<1\}\,? \]

Consider integral \[ I(s)=\int _C \frac {e^{s z^2}}{z^2-1}\,dz, \] \(C\) is the vertical line \(\operatorname {Re}z=2\), from \(z=2-i\infty \) to \(z=2+i\infty \), with large positive parameter \(s\). Find the three-term asymptotic expansion (approximation with three non-zero terms) of this integral as \(s\to +\infty \).