It seems the answers provided so far are missing the point. If we want to see a 4 quadrant atan2 version, that works for complex X and Y, can we use what Alpha suggests? Thus
atan2(U,V) = -i*log(U+i*V)./sqrt(U.^2 + V.^2)
I'll modify what @Stephen23 did to see what happens, for complex U and V. A serious problem is I would need to have a hyper-dimensional monitor to show the complete behavior. (My own personal holodeck is broken, awaiting parts. But shipping is very slow, coming from the future.) I would need to display essentially 6 dimensional behavior, and a regular monitor is just not up to snuff there. Nor are my eyes or my brain. So I can only do my best to look at what happens. fun = @(V,U) -1i*log((U+1i*V)./sqrt(U.^2+V.^2));
First, what happens for real U and V?
utest = randn(3);vtest = randn(3);
fun(vtest,utest)
ans =
1.8569 - 0.0000i 1.0516 + 0.0000i 1.4416 - 0.0000i
-0.0848 + 0.0000i -2.1910 + 0.0000i 2.0958 + 0.0000i
-2.7649 - 0.0000i 1.7415 + 0.0000i 1.0877 - 0.0000i
And that is clearly a problem, since for real U and V, we want the result to return a purely real result. We need to patch it for that case.
function Z = atan2alt(V,U)
Z = -1i*log((U+1i*V)./sqrt(U.^2+V.^2));
realInputs = (imag(U) == 0) & (imag(V) == 0);
Z(realInputs) = real(Z(realInputs));
atan2alt(vtest,utest)
ans =
1.8569 1.0516 1.4416
-0.0848 -2.1910 2.0958
-2.7649 1.7415 1.0877
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so atan2alt now returns purely real output for purely real inputs. Next, is it consistent with atan2?
norm(atan2(vtest,utest) - atan2alt(vtest,utest))
And that is (as we sometimes said in the deep past) good enough for government work. The difference is down into the realm of floating point trash. Now I am at least a little happy, since the two are consistent for real input.
Next is this a 4 quadrant result? Again, that will be difficult to show in detail. But perhaps we can do a little, by plotting the result along a simple path in the complex plane.
So u and v both now each live along lines in the complex plane, where the imaginary part is constant, and non-zero.
And of course, since atan2alt will now return a complex result, we need to look at both the real and imaginary part.
The result is a 4 quadrant version. Here we see the imaginary component is indeed non-zero.
Of course, this would need a great deal more thought, to see if there were additional problems other than the need for a real result. What happens when one or both of the arguments are zero? Again, unless the two are consistent, we have a problem.
atan2([0 1],[0;1])
ans =
0 1.5708
0 0.7854
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atan2alt([0 1],[0;1])
ans =
0 1.5708
0 0.7854
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Hey, that worked. I was actually just a little surprised. I was going to guess the code would fail when U==V==0, and perhaps return a NaN. But as it turns out, the result from the formula is then 0+infi, with an infinite imaginary part. But since it has a zero real part, the check at the end of atan2alt saved me.
I noticed that @M A wanted to see it work also for symbolic inputs. Again, I was wondering for just a second if it would survive here, but it did. Some might suggest plots I generated do not form a very strong test, since they presumed constant imaginary part. And, while it is not too difficult to conjure up a pair of paths for U and V that are more complicated, this seems a good point to stop. 'nuff said.
