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Euler's identity, sometimes called Euler's equation, is this equation:
- , pi
- , Euler's Number
- , imaginary unit
Euler's identity is named after the Swiss mathematician Leonard Euler. It is not clear that he invented it himself.
Respondents to a Physics World poll called the identity "the most profound mathematical statement ever written", "uncanny and sublime", "filled with cosmic beauty" and "mind-blowing".
Mathematical proof of Euler's Identity using Taylor Series[change | change source]
Many equations can be written as a series of terms added together. This is called a Taylor series
The Exponential function can be written as the Taylor series
As well, Sine can be written as
and Cosine as
Here, we see a pattern take form. seems to be a sum of sine and cosine's Taylor Series, except with all of the signs changed to positive. The identity we are actually proving is .
So, on the left side is , whose Taylor series is
We can see a pattern here, that every second term is i times sine's terms, and that the other terms are cosine's terms.
On the right side is , whose Taylor series is the Taylor series of cosine, plus i times the Taylor series of sine, which can be shown as:
if we add these together, we have
Now if we replace x with , we have..
Then we know that