Continuity

1 #complex/qual/work

Is the following function continuous, differentiable, continuously differentiable? f:R2Rf(x,y)={xyx2+y2(x,y)(0,0)0else.

? #complex/qual/work

Show that f(z)=z2 is uniformly continuous in any open disk |z|<R, where R>0 is fixed, but it is not uniformly continuous on C.

6 #complex/qual/work

Let F:R2R be continuously differentiable with F(0,0)=0 and F(0,0)<1.

Prove that there is some real number r>0 such that |F(x,y)|<r whenever (x,y)<r.

2 Multivariable derivatives #complex/qual/work

  • Complete this definition: “f:RnRm is real-differentiable a point pRn iff there exists a linear transformation…”

  • #complex/qual/work Give an example of a function f:R2R whose first-order partial derivatives exist everywhere but f is not differentiable at (0,0).

  • #complex/qual/work Give an example of a function f:R2R which is real-differentiable everywhere but nowhere complex-differentiable.

Implicit/Inverse Function Theorems

3 #complex/qual/work

Let f:R2R.

  • #complex/qual/work Define in terms of linear transformations what it means for f to be differentiable at a point (a,b)R2.

  • #complex/qual/work State a version of the inverse function theorem in this setting.

  • #complex/qual/work Identify R2 with C and give a necessary and sufficient condition for a real-differentiable function at (a,b) to be complex differentiable at the point a+ib.

5 #complex/qual/work

Let P=(1,3)R2 and define f(s,t):=ps36st+t2.

  • State the conclusion of the implicit function theorem concerning f(s,t)=0 when f is considered a function R2R.

  • State the above conclusion when f is considered a function C2C.

  • Use the implicit function theorem for a function R×R2R2 to prove (b).

There are various approaches: using the definition of the complex derivative, the Cauchy-Riemann equations, considering total derivatives, etc.

7 #complex/qual/work

State the most general version of the implicit function theorem for real functions and outline how it can be proved using the inverse function theorem.

Complex Differentiability

4 #complex/qual/work

Let f=u+iv be complex-differentiable with continuous partial derivatives at a point z=reiθ with r0. Show that ur=1rvθvr=1ruθ.

Tie’s Extra Questions: Fall 2016

Let u(x,y) be harmonic and have continuous partial derivatives of order three in an open disc of radius R>0.

  • Let two points (a,b),(x,y) in this disk be given. Show that the following integral is independent of the path in this disk joining these points: v(x,y)=x,ya,b(uydx+uxdy).

  • In parts:

  • Prove that u(x,y)+iv(x,y) is an analytic function in this disc.
  • Prove that v(x,y) is harmonic in this disc.

Tie’s Questions, Spring 2014: Polar Cauchy-Riemann #complex/qual/work

Let f=u+iv be differentiable (i.e. f(z) exists) with continuous partial derivatives at a point z=reiθ, r0. Show that ur=1rvθ,vr=1ruθ.

? #complex/qual/work

  • Show that the function u=u(x,y) given by u(x,y)=enyeny2n2sinnxfor nN is the solution on D={(x,y) |x2+y2<1} of the Cauchy problem for the Laplace equation 2ux2+2uy2=0,u(x,0)=0,uy(x,0)=sinnxn.

  • Show that there exist points (x,y)D such that lim sup.

#complex/qual/work