I used different codes to plot the phase of this function f(z) = (0.10 + 0.3i) + (0.4243 + 0.0017i)z + (0.9 − 0.001i)z^2 why I always find discontinuity at Z_R =0

I used the folloing 2 codes from Davidgoodman and Torsten (Thank you very much for all of them) respectivally to plot this function

f(z) = (0.10 + 0.3i) + (0.4243 + 0.0017i)z + (0.9 − 0.001i)z^2

My question IS:

why always find dis continuity at Z_R (the real part of Z) in the phase of this function? and how to avoid this discontinuity?

this are the both codes

This code by Davidgoodman (Thank you very much)

re_z = -1:0.01:1;

im_z = -1:0.01:1;

[re_z,im_z] = meshgrid(re_z,im_z);

z = re_z + 1i*im_z;

caxis([-pi pi])

xlabel('x')

ylabel('y')

p=[(0.9-0.001i) (0.4243 + 0.0017i) (0.10 + 0.3i)];

f_of_z_result = polyval(p,z);

p1=[(0.9 + 0.001i) (0.2121 - 0.0008i) (0.10 -0.3i)];

f_of_1_over_z_result = polyval(p1,1./z);

figure();

subplot(2,2,1)

surf(re_z,im_z,abs(f_of_z_result),'EdgeColor','none')

title('|f(z)|')

xlabel('Z_R')

ylabel('Z_I')

% Zlabel('|f(z)|')

grid on

subplot(2,2,2)

surf(re_z,im_z,angle(f_of_z_result),'EdgeColor','none')

title('phase of f(z)')

xlabel('Z_R')

ylabel('Z_I')

subplot(2,2,3)

surf(re_z,im_z,abs(f_of_1_over_z_result),'EdgeColor','none')

title('|f(1/z)|')

xlabel('Z_R')

ylabel('Z_I')

subplot(2,2,4)

surf(re_z,im_z,angle(f_of_1_over_z_result),'EdgeColor','none')

title('phase of f(1/z)')

xlabel('Z_R')

ylabel('Z_I')

% Zlabel('phase of f(z)')

grid on

This code by Torsten (Thank you very much)

f = @(x,y) (0.3162).*exp(1i*((0.398)*pi))+((x+1i*y).*(0.4243).*exp(1i*((0.0013)*pi)) + (x+1i*y).^2*(0.9).*exp(-1i*(0.00035)*pi));

r = 0:0.01:0.9;

phi = 0:0.01:2*pi;

[R,PHI] = meshgrid(r,phi);

X = R.*cos(PHI);

Y = R.*sin(PHI);

F=f(X,Y);

figure

subplot(2,2,1)

surf(X,Y,abs(f(X,Y)),'EdgeColor','none')

view(2)

colorbar

title('|f(z)|')

xlabel('Z_R')

ylabel('Z_I')

% Zlabel('|f(z)|')

grid on

subplot(2,2,2)

surf(X,Y,angle(f(X,Y)),'EdgeColor','none')

view(2)

colorbar

title('phase of f(z)')

xlabel('Z_R')

ylabel('Z_I')

% Zlabel('|f(z)|')

grid on

subplot(2,2,3)

surf(X,Y,angle(f(X,Y)),'EdgeColor','none')

view(2)

colorbar

title('phase of f(z),view(2)')

xlabel('Z_R')

ylabel('Z_I')

% Zlabel('|f(z)|')

grid on

As we can see both codes work with |f(z)|, but not with the phase of the function.

I appriciate any help.

[Torsten]

Taking the phase of an even holomorphic function does not need to be continous.

I tried to explain it to you by the simple function f(z) = z, but you don't seem to have enough knowledge about complex analysis to understand my answer.

Read a book about complex analysis, e.g. Chapter 10 - 15 from

https://59clc.files.wordpress.com/2011/01/real-and-complex-analysis.pdf

or in a more basic fashion

https://ryr2008.files.wordpress.com/2015/09/marsden-jerrold-michael-j-hoffman-basic-complex-analysis.pdf

[Jeffrey Clark]

@Aisha Mohamed, when looking at phase (angle) over some extent of a function it is visually easier to see relationships when unwrap is used to make it linear continuous: Shift phase angles - MATLAB unwrap (mathworks.com)