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While studying complex numbers, I came across topics like:

  1. Sums of series of complex numbers
  2. Nth roots of complex numbers and so on...

However, I haven't actually found any 'real life' uses of them. I was going to ask my teacher, but thought that here may be some experts on complex numbers, who would be able to give me an insight into the use of them.

I would be really grateful if someone could enlighten me. I know that complex numbers are generally used a lot, but what about this specific topic of sums of series of them?

Nikolai
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  • What exactly do you mean by "sums of series" ? Do you mean something like $$\sum_{k=1}^n\sum_{l=1}^\infty z_{k,l},$$ where each $z_{k,l}$ is a complex number? – Maximilian Janisch Nov 05 '21 at 15:01
  • Have you seen this MSE post? – MyMolecules Nov 05 '21 at 15:04
  • Analytic Functions can be represented by sums of complex numbers. They are very important for numerical approximations to solutions of certain differential equations including problems that arise in electromagnetism and quantum mechanics. In the case of EM, such terms are the only form in which I've encountered an analytic representatio of fringe fields. Finite sums are at the heart of the cubic formula for solving cubic equations. Frequently proofs can be simplified by re-expressing relevant terms as complex numbers. – TurlocTheRed Nov 05 '21 at 15:44

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Take a look at the topic of Fourier series, particularly Fourier series with complex coefficients and complex values. There are many, many, many applications of Fourier series in science and technology; that wikipedia link gives a brief list of examples.

Lee Mosher
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