today in class i was asked what was wrong with this
e^ipi + 1=0
e^ipi=-1
e^2ipi=1
2ipi=ln1
this is obviously wrong, is it the stage where you square e^ipi,
i havnt come across complex powers before. thanks for any help sorry if this is a stupid question.
Hello hmmmm,
See here: Euler's identity - Wikipedia, the free encyclopedia
No, and that's true
The problem is when you take the logarithm.
Because where k is an integer, this is why your step is wrong.
So following this reasoning, you can have which is indeed true.
You can have a look at this : Complex logarithm - Wikipedia, the free encyclopedia
Imagine that (this is a definition)
Since the cosine and the sine are -periodic functions, we can write :
The logarithm in the complex values has an infinity of values, unless you define a continuous logarithm (but I don't think this is something you need to know yet).
another way of seeing it is that
So you can multiply by , which gives , it won't change the value.