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The real cosine function can be extended using the exponential function:
As a power series the equivalent definition is:
This series converges everywhere in the complex plane. If we increase the degree of the Taylor polynomial, this polynomial approaches the function more and more. We can see it if we look at the Remainder (the difference between the function and the polynomial):
The Cosine Function is periodic with period
REFERENCES
Tristan Needham - Visual Complex Analysis. (pags. 84) - Oxford University Press
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