State the Cauchy-Riemann equations.
Define what it means for a function to be
Holomorphic
Meromorphic
Analytic
Harmonic
Uniformly continuous
Uniformly bounded
Entire
What does it mean for a sequence or series to uniformly converge?
State the Laplace equation.
What is the Dirichlet problem?
Discuss how to carry out partial fraction decomposition
Determine the radius of convergence of the power series for \(\sqrt z\) expanded at \(z_0= 4 + 3i\).
What is the logarithmic derivative?
Find a function \(f\) such that \(f^2\) is analytic on the open unit disc but \(f\) is not.
Show that \(f(z)=z^2\) is uniformly continuous on every disk \(|z|<R\), but is not uniformly continuous on \(\CC\).
- Use that Lipschitz implies uniformly continuous.
Identify \(\RR^2\) with \(\CC\) and give a necessary and sufficient condition for a real-differentiable function at \((a,b)\) to be complex differentiable at \(a+ib\).
- Derivation of CR equations: approach along totally real and totally imaginary paths for \(h\).
Let \(f=u+iv\) be complex differentiable with continuous partial derivatives at \(z=re^{i\theta}\), \(r\ne0\). Derive the polar Cauchy–Riemann equations.
- Polar coordinates, chain rule, Cauchy-Riemann equations.
Suppose \(f(z)=\sum_{n\ge0}a_nz^n\) has radius of convergence exactly \(1\). Give an example whose series converges at every point of \(S^1\); give an example for which \(f\) is analytic at \(1\) but \(\sum a_n\) diverges; and prove that \(f\) cannot be analytic at every point of \(S^1\).
- Converges everywhere on \(S^1\): take \(\sum z^k/k^2\).
Find the Laurent expansions of \[ {z+1\over z(z-1)^2} \] about \(z=0\) and \(z=1\).
Solution
The Cartesian Cauchy–Riemann equations for \(f=u+iv\) are \[ u_x=v_y,\qquad u_y=-v_x. \] A function on an open set is holomorphic if it is complex differentiable at every point; meromorphic if it is holomorphic except for isolated poles; analytic if it is locally represented by a convergent power series (equivalently, in one complex variable, holomorphic); harmonic if it is \(C^2\) and satisfies \(u_{xx}+u_{yy}=0\); entire if it is holomorphic on all of \(\CC\). Uniform continuity means that for every \(\epsilon>0\) there is a \(\delta>0\) working simultaneously for all pairs of points in the domain; uniform boundedness means that one constant \(M\) bounds the modulus everywhere. A sequence \(f_n\to f\) converges uniformly if \(\sup_x|f_n(x)-f(x)|\to0\); a series converges uniformly when its partial sums do. The Laplace equation is \(u_{xx}+u_{yy}=0\). The Dirichlet problem asks for a harmonic function on a domain with prescribed boundary values.
For partial fractions, factor the denominator over \(\CC\) and write one term \(A_{j,k}/(z-a_j)^k\) for every multiplicity \(k\) of every pole \(a_j\); multiplying through by the denominator and comparing coefficients (or evaluating derivatives at the poles) determines the constants.
A local branch of \(\sqrt z\) about \(z_0=4+3i\) has radius of convergence \[ \boxed{5}. \] Indeed, the open disk of radius \(5\) about \(z_0\) avoids \(0\) and is simply connected, so it admits a holomorphic square root. A larger Taylor disk would contain \(0\); if a holomorphic \(g\) there satisfied \(g^2=z\), then \(g(0)=0\) and differentiating would give \(2g(0)g'(0)=1\), impossible.
The logarithmic derivative of a nonzero holomorphic function is \(f'/f\). An example with \(f^2\) analytic but \(f\) not analytic is \[ f(z)=\begin{cases}1,&\Im z\ge0,\\-1,&\Im z<0, \end{cases} \qquad f^2\equiv1. \]
For \(f(z)=z^2\) and \(|z|,|w|<R\), \[ |z^2-w^2|\le2R|z-w|, \] so \(z^2\) is uniformly continuous on that disk. On \(\CC\), take \(z_n=n\) and \(w_n=n+1/n\); then \(|z_n-w_n|\to0\) but \(|z_n^2-w_n^2|\to2\), so uniform continuity fails.
If a real-differentiable map \(f=u+iv\) is given at \((a,b)\), then it is complex differentiable there exactly when its real derivative is \(\CC\)-linear, equivalently exactly when the Cauchy–Riemann equations hold at \((a,b)\).
In polar coordinates, the chain rule gives \[ u_r=u_x\cos\theta+u_y\sin\theta, \qquad v_\theta=r(-v_x\sin\theta+v_y\cos\theta). \] Using \(u_x=v_y\) and \(u_y=-v_x\) yields \[ u_r={1\over r}v_\theta, \qquad v_r=-{1\over r}u_\theta. \]
For the radius-one power-series questions:
- \(\sum_{n\ge1}z^n/n^2\) has radius \(1\) and converges absolutely at every point of \(S^1\).
- \(f(z)=1/(1+z)=\sum_{n\ge0}(-1)^nz^n\) has radius \(1\) and is analytic at \(1\), but \(\sum_{n\ge0}(-1)^n\) diverges.
- If \(f\) were analytic at every point of \(S^1\), compactness of \(S^1\) would give finitely many neighborhoods on which \(f\) is analytic, together containing an annulus around \(S^1\). Joined with the unit disk, these give analyticity on a disk \(|z|<1+\epsilon\) for some \(\epsilon>0\), contradicting the assumed Taylor radius \(1\).
Finally, \[ {z+1\over z(z-1)^2}={1\over z}-{1\over z-1}+{2\over(z-1)^2}. \] About \(0\) this gives \[ {1\over z}+\sum_{n\ge0}(2n+3)z^n, \qquad 0<|z|<1, \] and \[ \sum_{n\ge2}(2n-3)z^{-n}, \qquad |z|>1. \] Writing \(w=z-1\), the expansions about \(1\) are \[ {2\over w^2}-{1\over w}+\sum_{n\ge0}(-1)^nw^n, \qquad 0<|w|<1, \] and \[ {1\over w^2}+\sum_{n\ge3}(-1)^{n-1}w^{-n}, \qquad |w|>1. \]