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I'm learning some math by myself and I thought of this question that has to do with how useful complex numbers are:

I'm looking for a specific example of an equation that has the following properties:

  • The equation is true, i.e., its LHS and RHS are always equal, no matter the values of its variables, if any.
  • The LHS and the RHS are algebraic expressions (no limits, derivatives, integrals, sin, cos, etc...)
  • The equation only involves real numbers.
  • Its shortest equality proof involves complex numbers.
  • The proof ends with no complex numbers.
  • It has a longer equality proof that doesn't involve complex numbers.
  • It's a short equation (this property is subjective).

Please, let me know if my question is wrong or if no such equation can exist.

Otakar Molnár López
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    Similar question (but probably different enough from this one to not be a duplicate): https://math.stackexchange.com/questions/2075039 – JimmyK4542 May 18 '23 at 02:28
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    The question is overly broad There are too many proofs to count in geometry, trigonometry, calculus. etc, where neither the premise nor the conclusion involve complex numbers, but the shortest and most direct/elegant proof uses complex numbers. A few examples at random 1, 2, 3. – dxiv May 18 '23 at 03:33
  • I edited the question to reflect more faithfully what I have in mind. – Otakar Molnár López May 19 '23 at 19:37

1 Answers1

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The following equality is generally proved with complex analysis:

$$\int_{-\infty}^{\infty} \frac{\cos(tx)}{x^2+1}dx=\pi e^{-|t|}$$

(for $t\in\mathbb{R}$). It's a neat application of the residue theorem where a contour is either taken above or below the axis depending on whether $t>0$.

QC_QAOA
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