2*pi*i = 0? euler's formula

  • Thread starter tom92373
  • Start date
  • #1
3
0
I was playing around with euler's formula the other day and found this odd proof which says 2ipi=0. I know this is obviously wrong, but what was the step that went wrong? The "proof" goes like this:

start with e^(i*pi)+1=0
so -e^(i*pi)=1
multiply both sides by e e(-e^(i*pi))=e
factor a (-1) (-1)(e)(e^(i*pi))=e
add exponents (-1)(e^(i*pi+1))=e
take ln of both sides ln((-1)(e^(i*pi+1)))=1
use properties of logs ln(-1) + ln(e^(i*pi+1))=1
take ln of -1 i*pi + ln(e^(i*pi+1))=1
properties of logs i*pi + (i*pi+1)ln(e)=1
take ln(e) i*pi + i*pi+1=1
subtract 1 i*pi + i*pi=0
combine like terms 2*pi*i=0 ...

this even works in the original formula:
e^(i*2*pi)=cos(2*pi)+isin(2*pi)
e^(i*2*pi)=1
e^(i*2*pi)=e^0
i*2*pi=0 ...
 

Answers and Replies

  • #2
623
0


Well what's the period of sine and cosine? What's e^(4pi*i)?
 
  • #3
1
0


in complex analysis, log function is a Multi-valued function,ln(z) = |z| + arg(z), you should choose a single value branch for log, or use Riemann surface
 
  • #4
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
132


e^(i*2*pi)=e^0
i*2*pi=0

That inference, from line 1 to line 2 is invalid.
 
  • #5
HallsofIvy
Science Advisor
Homework Helper
41,833
956


In "polar form", [itex]z=re^{i \theta}[/itex], [itex]\theta[/itex] represents the angle the line from 0 to z makes with the positive real axis. In that sense, yes, [itex]2\pi[/itex] is the same angle as 0. That's why, as arildno suggested, in complex numbers, the exponential function is no longer single valued. f(x)= f(y) implies x= y only if f is single valued.
 
  • #6
3
0


Ok I get it; so it's because complex exponentials have more than one solution.
 
  • #7
534
1


That's why, as arildno suggested, in complex numbers, the exponential function is no longer single valued. f(x)= f(y) implies x= y only if f is single valued.
I think you mean it's not one-to-one! It's the logarithm that's not single-valued. :)
 

Related Threads on 2*pi*i = 0? euler's formula

Replies
7
Views
3K
  • Last Post
Replies
9
Views
4K
  • Last Post
Replies
3
Views
733
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
10
Views
5K
  • Last Post
2
Replies
37
Views
6K
  • Last Post
Replies
11
Views
2K
  • Last Post
Replies
6
Views
3K
  • Last Post
Replies
1
Views
2K
Top