MATH529

SP.22: Homework 11

Instructions

The assignment consists solely of exercises. The model solutions show verifications through Mathematica. Though beneficial, the Mathematica verifications are not required.

Exercises

Solve 4 questions of your choice from each of the following subsections:  17.1 (p.834), 17.2 (p.838),  17.4 (p. 846), 17.5 (p. 850)

Clear["Global`*"];

Model solution 17.1.1

Compute 2(i^3)-3(i^2)+5i.
Solution: 2(i^3)-3(i^2)+5i=2 (-i)-3 (-1)+ 5 i=3+3i
Verify:

2(I^3)-3 (I^2)+5 I

Model solution 17.2.5

Write z=1+i in polar form
Solution: z=r (e^(i θ))=a+ i b, r=(Sqrt[(a^2)+(b^2)])=(Sqrt[1+1])=(Sqrt[2]), θ=atan(b/a)=atan(1)=π/4, z=(Sqrt[2]) (e^(i π/4)) 
Verify:

z=1+I; {Abs[z],Arg[z]}

Model solution 17.4.15

Evaluate f(z)=2x-(y^2)+i(x (y^3)-2(x^2)+1) at z=2i
Solution: z=x+i y=2i⇒x=0, y=2⇒, f(2i)=0-(2^2)+i(0-0+1)=-4+i,
Verify:

f[z_]=2 Re[z] - Im[z]+I( Re[z]((Im[z])^3)-2 ((Re[z])^2)+1); f[2I]

Model solution 17.5.3

Show that f(z)=Re(z) is not analytic at any point
Solution: z=x+i y⇒f(z)=x=u(x,y)+i v(x,y), Cauchy-Riemann conditions are not verified
(Subscript[u, x])=1≠0=(Subscript[v, y]), hence f not analytic
Verify:

u[x_,y_]=x; v[x_,y_]=0; CR = {D[u[x,y],x]==D[v[x,y],y],D[u[x,y],y]==-D[v[x,y],x]}

Problems

There are no problems in this assignment.

Projects

There are no projects in this assignment.