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Homework 4 (MATH 1199 Fall 2019)
1. Write an ϵ-δ proof that the following functions are continuous.
(a) {f:CCf(z)=3z7
(b) {f:CCf(z)=7
(c) {f:CCf(z)=Re(z)


2. Use the limit definition of the derivative to compute
(a) ddz2z2+z
(b) ddzz35z

3. Verify that the Cauchy-Riemann equations hold for the given function.
(a) {f:CCf(z)=3z2+1
(b) {f:CCf(z)=z513z4+3z210
(c) {f:C{0}Cf(z)=1z
(d) {f:CCf(x+iy)=exeiy,x,yR

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:CCf(z)=z+¯z
(b) {f:CCf(x+iy)=exeiy,x,yR

5. Find...
(a) exp(7+9πi)
(b) exp(17+πi4)
(c) Log(ei)
(d) Log(1+i)
(e) log(3i) (note: lowercase log here!! "multivalued"!!!)
(f) log(1i)