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Complex sine

The complex sine can be expressed with exponential functions as

 


Explanation

We start with the power series for the complex sine

and multiply all terms with 2i and get

By conversion with the imaginary unit all terms obtain a plus sign

You can also write it as

We add even exponents and subtract these immediately

Rearranging gives

In brackets are two power series for exponential functions, and thus

so that

 


Example 1

You can see that sin (0) = 0, as

 


Additional information

This formula for the sine was described by the Swiss mathematician Leonhard Euler.

 


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