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
HistoryThis formula for the sine was described by the Swiss mathematician Leonhard Euler (1707 - 1783). |