Euler's Identity

Question

Can anyone explain where did I actually commit a mistake? In the end I got the result that e^(2\pi )=1 (which is clearly not true)

$a^{b}$ is not uniquely defined number for $a,b \in \mathbb C$ and your manipulations are invalid. Until you are familiar with Complex Analysis you should use $a^{b}$ only for $a>0$. — Kavi Rama Murthy, Dec 25 '19 at 09:37

score 1 · Answer 1 · answered Dec 25 '19 at 10:16

1

The trick comes when you take the non-integer power of a negative number or a complex number with a non-zero imaginary part as Kavi mentioned. A contradiction involving exponents

answered Dec 25 '19 at 10:16

Alessio K

10,599

score 0 · Answer 2 · answered Dec 25 '19 at 10:21

0

You can refer to the following paradox: $$(1) ^{0.5}=(-1\times -1) ^{0.5}=(-1) ^{0.5}\times (-1) ^{0.5}=i\times i=-1$$. (Since $x^y$ is not well defined for $x, y\in\mathbb{C}$

answered Dec 25 '19 at 10:21

Drinzjeng Triang

815

Euler's Identity

2 Answers2