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Darío A. Gutiérrez


My favorite Zitate

-"Sir, an equation has no meaning for me unless it expresses a thought of God." (Srinivasa Ramanujan)

-"Denn die Mathematik ist es, die uns vor dem Trug der Sinne schützt und die uns den Unterschied zwischen Schein und Wahrheit kennen lehrt.." (Leonhard Euler)


My favorite Identity
\begin{align} e &= 2,71828182845904523… \\ \pi &= 3,14159265358979323… \\ i&=\sqrt{-1}\\ \end{align}
\begin{align} e^{ix} &= 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \cdots \\ &= 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \cdots \\ &= \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \right) + i\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \right) \\ &= \cos x + i\sin x \\ &= 1 \end{align}

$$e^{i\pi} = 1$$ enter image description here




My favorite Answers on Mathematics StackExchange:

How can I calculate $\alpha=\arccos\left(-\frac{1}{4}\right)$ without using a calculator?

Compute : $\int\frac{x+2}{\sqrt{x^2+5x}+6}~dx$

Prove that $\sum_{k=0}^nk{m+k \choose m}=n{m+n+1\choose m+1}-{m+n+1 \choose m+2}$

Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ ?


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