what dose z represent in this graph?

I am studying these graphs , and I really confused in some points here.

1- when we take arbitrary point in this figure for example as we can see in the first figure (x= -0.465, y=0.325, z=0.0084), we see that z>0 (greter than 0)

but when we calculate (cos(angle(-0.465+0.325i))) , we got it = -.8196 <0 (les than zero ))

it is clear from this figure z does not equal the value of the function (cos(angle(-0.465+0.325i))), and

if z indicate to the sign of this function , why we got z>0 whereas the value of this function <o?

My questions are,

Is z in this figure indiacet to the value of (cos(angle(-0.465+0.325i))) or only tells us where (cos(angle(-0.465+0.325i))) positinve and negative? and why we find this contrast?

I will appreciate any help.

[dpb]

We don't know what created the figures -- it may not be what the title says it is, first off...

x=linspace(-2*pi,2*pi);
[X,Y]=meshgrid(x,x);
Z=cos(angle(complex(X,Y)));
figure
hIm=imagesc(x,x,Z);
colorbar
datatip(hIm,-0.465,0.325)

results in the following image

The exact point isn't in the X,Y grid, but is close by and we see that the result of the given functional actually is pretty close to the value given of ~-0.8.

Hence, we're left to conclude that while the image has a similar functional shape, with the diagonal rays, it isn't the same functional plotted as the equation you've written. For one thing, it is concave upwards to both -x and +x directions (yellow on both LH and RH axes) and the apex isn't at the origin but the LH/RH sides are shifted up/down, respectively. There's been some other transform done to create the image more.

[Walter Roberson]

Looks okay to me.

format long g

ZR = linspace(-8,8,500);
ZI = linspace(-8,8,500).';
Z = cos(angle(ZR + ZI*1i));
h = pcolor(ZR, ZI, Z);
h.EdgeColor = 'none';
colorbar;

xt = -0.465; yt = 0.325;
[~, cidx] = min( abs(ZR-xt) ); closest_x = ZR(cidx);
[~, ridx] = min( abs(ZI-yt) ); closest_y = ZI(ridx);

zthere = Z(ridx, cidx)

zthere = 
        -0.809942121543021

[xt,yt, cos(angle(xt+yt*1i))]

ans = 1×3

-0.465 0.325 -0.819645759106337

[closest_x, closest_y, cos(angle(closest_x + closest_y*1i))]

ans = 1×3

-0.464929859719439 0.336673346693387 -0.809942121543021

[dpb]

@Walter Roberson -- yes, that duplicates my but these don't match the OP's original image nor its datatip. His image doesn't match the title/functional form he thinks it's supposed to be/it has been titled as being.

[Walter Roberson]

@dpb

My answer was in preparation when you posted yours. We happened to close to the same thing.

[Torsten]

Maybe I'm blind, but where do you get from that f_k(z) = z ?