Loading [MathJax]/jax/output/HTML-CSS/jax.js
Back to the class
Homework 4 (MATH 1199 Fall 2019)
1. Write an ϵ-δ proof that the following functions are continuous.
(a) {f:C→Cf(z)=3z−7
(b) {f:C→Cf(z)=7
(c) {f:C→Cf(z)=Re(z)
2. Use the limit definition of the derivative to compute
(a) ddz2z2+z
(b) ddzz3−5z
3. Verify that the Cauchy-Riemann equations hold for the given function.
(a) {f:C→Cf(z)=3z2+1
(b) {f:C→Cf(z)=z5−13z4+3z2−10
(c) {f:C∖{0}→Cf(z)=1z
(d) {f:C→Cf(x+iy)=exeiy,x,y∈R
4. Explain why the function is not differentiable everywhere using an appropriate calculation (i.e. the Corollary on p.31 of the notes).
(a) {f:C→Cf(z)=z+¯z
(b) {f:C→Cf(x+iy)=exe−iy,x,y∈R
5. Find...
(a) exp(−7+9πi)
(b) exp(−17+πi4)
(c) Log(−ei)
(d) Log(1+i)
(e) log(√3−i) (note: lowercase log here!! "multivalued"!!!)
(f) log(1−i)