The English used in this article or section may not be easy for everybody to understand. (April 2012)
It features the following mathematical constants:
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.
As well, the sine function 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:
Since we know that and , we have:
which is the statement of Euler's identity.
Related pages[change | change source]
References[change | change source]
- "Euler's Formula: A Complete Guide — Euler's Identity". Math Vault. 2020-09-30. Retrieved 2020-10-02.
- Weisstein, Eric W. "Euler Formula". mathworld.wolfram.com. Retrieved 2020-10-02.
- Hogenboom, Melissa. "The most beautiful equation is... Euler's identity". www.bbc.com. Retrieved 2020-10-02.
- Sandifer, C. Edward 2007. Euler's greatest hits. Mathematical Association of America, p. 4. ISBN 978-0-88385-563-8
- Crease, Robert P. (2004-10-06). "The greatest equations ever". IOP. Retrieved 2016-02-20.