# How can the following mathematical statements be proven?

I have these two mathematical statements: 1) $e^{i\pi}=-1$ and 2) $\ln(-1)=i\pi$. Now I want a proof of these statements. Can anyone help me proving these statements?

• Are you familiar with Euler's identity? How have you defined the functions involved, and what have you tried? – user61527 Nov 23 '13 at 21:26
• Do you know the definitions of the functions involved? – Git Gud Nov 23 '13 at 21:28
• You can prove $e^{ix}=\cos(x)+i\sin(x)$ by deriving both sides, then put $x=\pi$ – LeeNeverGup Nov 23 '13 at 21:28
• It depends on how $e^z$ was defined to you. – Emanuele Paolini Nov 23 '13 at 21:28
• @NorbertWillhelm, it's time for you (and us all) to learn the lesson: WA is wrong lots of times in lots of different subjects. It is, of course, a great site where one can verify what was already done, but to base one's answer on it...tsk,tsk,tsk. Fly by Night is right, and the reason is a little deep if you haven't yet studied complex analysis. – DonAntonio Nov 24 '13 at 12:20

That's straightforward $$e^{i\pi}=\cos(\pi)+i\sin(\pi)=-1$$
• You can put MathJaX code between $$. – Ian Mateus Nov 23 '13 at 21:35 • @IanMateus thank you :) I didn't realize that... this is my first answer here ... :D – Jekyll Nov 23 '13 at 21:42 If you take the definition (high school, usually)$$\text{For}\;\;x\in\Bbb R\;,\;\;e^{ix}:=\cos x+i\sin x\implies e^{\pi i}=\cos\pi+i\sin\pi=-1$$and now, choosing the branch \;\{z\in\Bbb C\;;\;\text{Im}\,z\ge 0\}\; for the logarithm (in this case, it means that \;\arg(-1)=\pi\; ) , we get$$\text{Log}\,(-1):=\log|-1|+i\arg(-1)=0+\pi i