Imaginary number challenge (3)

Algebra Level 4


If the principal value of the expression above can be written in the form \(\exp\left(\dfrac{x!\pi}y\right)\) where \(x\) and \(y\) are positive integers, find \(x+y\).

Details and Assumptions:

  • Every complex number \(z\) can be written in the form \(z=re^{i\theta+2k\pi}\) for some real \(r,\theta\) and all \(k\) such that \(-\pi\lt \theta\leq\pi\). The value when \(k=0\) is called the principal value of \(z\).
  • \(i=\sqrt{-1}\) denotes the imaginary unit.
  • \(\exp(x)=e^x\) where \(e\) denotes the Euler's number.
  • \(n!\) denotes the factorial function. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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