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I know the very broad applications of complex numbers in engineering, although I don't understand the applications of complex functions of complex variables. Things such as the Residues, Cauchy Goursat, Cauchy integral formula are being used in advanced topics in Theoretical Physics. How about Electrical & Electronics Engineering?

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  • $\begingroup$ What about the reverse question: are they wasting your time by including a course you don’t need? $\endgroup$
    – Solar Mike
    Commented Nov 19, 2023 at 15:54
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    $\begingroup$ One has to recognize that any effective higher education HAS to over educate you. So even if it does not particularily matter for your day to day job it might be a essential tool in some subfield research. So generally i dont need even a quarter of all the things they taught me. But its a good foundation to things they didnt teach me. $\endgroup$
    – joojaa
    Commented Nov 20, 2023 at 21:08
  • $\begingroup$ All EEs at my university have to take a mechanical dynamics course in second year. It is by far the most difficult of our courses in that year. Our professor told us that one time, a student asked him why they had to take this course and his answer was because motors spin. So from that one three term equation $$v = \omega r$$ on the inside cover of our textbook which we need, we take the entire mechanical dynamics course. Well, it's helped me in other ways too such as calculating motor torque requirements for land vehicles on different terrain, inclines, accelerations, speeds, wheel diameters. $\endgroup$
    – DKNguyen
    Commented Nov 22, 2023 at 22:06
  • $\begingroup$ The future is unknown, so they prepare you with knowledge you may need. It is easier to accept the brainwashing of education than fight it! $\endgroup$ Commented Nov 27, 2023 at 3:03

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The obvious examples would be the solutions to Maxwells equations for wave guides.

Post WWII, MIT academics formed the opinion that Electrical and Electronics engineering was changing so fast that nothing practical they taught their students would be of any enduring value in their careers. The solution that was adopted was that professional engineering students should be taught underlying theory, which would remain true and provide the basis for ongoing continuing practice. There was never the expectation that common practicing EE's would actually use the Cauchy residue theorem to solve electromagnetic wave equations on a daily basis: just that they would be able to understand the derivation and apply the results correctly.

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Complex numbers come up in electronics or electrical engineering as soon as AC voltages meet inductors or capacitors. With an inductive or capacitive load, the current ends up out of phase with the voltage. A complex number can be treated as a phase and magnitude, just what you need to handle this situation.

So the impedance of a device to AC is the equivalent of the resistance to DC, but it's a complex number. And the result of applying Ohm's law also ends up complex.

How complex this complex maths needs to be will depend a lot on what you're studying.

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These are really the basics of ONE-variable complex analysis (so complex variable, not complex variables). Of course you need to know the stuff - going beyond the first two years of physics will be hard if you don't. (In fact, once you know more analysis, the way even very good introductory physics textbooks are written starts looking very funny in retrospect.)

More concretely: it should be pretty clear that an electrical engineer should have a solid grounding in Fourier analysis. Going beyond a certain basic level in Fourier analysis without knowing any complex analysis is... rather awkward.

(I was just wondering: how do you prove that the Gaussian is its own Fourier transform without using complex analysis? I looked it up, and yes, there is a short proof without complex analysis, but it must have been found by someone clever. With complex analysis, the proof is extremely natural.)

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