Sizler de bildiginiz güzel özdeşlikleri yazınız.
1."Logaritma"
$$log(1+2+3)=log1+log2+log3$$
$$\sqrt{n^{\log n}}=n^{\log \sqrt{n}}$$
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2."Toplamlara dair"
$$\left(\sum\limits_{k=1}^n k\right)^2=\sum\limits_{k=1}^nk^3 $$
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3."Gamma fonksiyonundan bir çıkarım"
$$\infty! = \sqrt{2 \pi}$$
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4."https://ocw.mit.edu/courses/mathematics/18-312-algebraic-combinatorics-spring-2009/readings-and-lecture-notes/MIT18_312S09_lec10_Patitio.pdf"
$$(1+x)(1+x^{2})(1+x^{3}) \cdots = \frac{1}{(1-x)(1-x^{3})(1-x^{5}) \cdots}$$
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5."Machin's Formulü:"
$$\begin{eqnarray}\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}\end{eqnarray}$$
"Trigonometri"
$$\sec^2(x)+\csc^2(x)=\sec^2(x)\csc^2(x)$$
$$\frac{1}{\sin(2\pi/7)} + \frac{1}{\sin(3\pi/7)} = \frac{1}{\sin(\pi/7)}$$
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
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6."$\pi$"
$$\frac{\pi}{2} = \frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\frac{8}{7}\cdot\ldots\\$$
$$\frac{\pi}{4} = 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\ldots\\$$
$$\frac{\pi^2}{6} = 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\ldots\\$$
$$\frac{\pi^3}{32} = 1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+\frac{1}{9^3}+\ldots\\$$
$$\frac{\pi^4}{90} = 1+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}+\frac{1}{5^4}+\ldots\\$$
$$\frac{2}{\pi} = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdot\ldots\\ $$
$$\pi = \cfrac{4}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\cfrac{4^2}{9+\ldots}}}}}\\$$
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7.
$$\begin{eqnarray}1^{3} + 2^{3} + 2^{3} + 2^{3} + 4^{3} + 4^{3} + 4^{3} + 8^{3} = (1 + 2 + 2 + 2 + 4 + 4 + 4 + 8)^{2}\end{eqnarray}$$
$$\large{1,741,725 = 1^7 + 7^7 + 4^7 + 1^7 + 7^7 + 2^7 + 5^7}$$
$$\large{111,111,111 \times 111,111,111 = 12,345,678,987,654,321}$$
$$\large{1,741,725 = 1^7 + 7^7 + 4^7 + 1^7 + 7^7 + 2^7 + 5^7}$$
$$10^2+11^2+12^2=13^2+14^2$$
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8.
$$\displaystyle\big(a^2+b^2\big)\cdot\big(c^2+d^2\big)=\big(ac \mp bd\big)^2+\big(ad \pm bc\big)^2$$
9.
$$\begin{eqnarray}\sum_{i_1 = 0}^{n-k} \, \sum_{i_2 = 0}^{n-k-i_1} \cdots \sum_{i_k = 0}^{n-k-i_1 - \cdots - i_{k-1}} 1 = \binom{n}{k}\end{eqnarray}$$
10.
$$\frac{e}{2} = \left(\frac{2}{1}\right)^{1/2}\left(\frac{2\cdot 4}{3\cdot 3}\right)^{1/4}\left(\frac{4\cdot 6\cdot 6\cdot 8}{5\cdot 5\cdot 7\cdot 7}\right)^{1/8}\left(\frac{8\cdot 10\cdot 10\cdot 12\cdot 12\cdot 14\cdot 14\cdot 16}{9\cdot 9\cdot 11\cdot 11\cdot 13\cdot 13\cdot 15\cdot 15}\right)^{1/16}\cdots$$
11.
$$\int_0^\infty\sin\;x\quad\mathrm{d}x=1$$
$$\int_0^\infty\ln\;x\;\sin\;x\quad \mathrm{d}x=-\gamma$$
12."M.V Subbarao özdeşliği: $n(\in\mathbb Z)>22$ ve asal olsun"
$$n\sigma(n)\equiv 2 \pmod {\phi(n)}$$
13.
$$i^i = \exp\left(-\frac{\pi}{2}\right)$$
$$\root i \of i = \exp\left(\frac{\pi}{2}\right) $$
$$\Rightarrow$$
$$\int_{0}^{\infty }\cos\left ( 2x \right )\prod_{n=0}^{\infty}\cos\left ( \frac{x}{n} \right )~\mathrm dx\approx \frac{\pi}{8}-7.41\times 10^{-43}$$
15.$$\int_0^\infty\frac1{1+x^2}\cdot\frac1{1+x^\pi}dx=\int_0^\infty\frac1{1+x^2}\cdot\frac1{1+x^e}dx$$
16.
$$\begin{eqnarray}\sum_{k = 0}^{\lfloor q - q/p) \rfloor} \left \lfloor \frac{p(q - k)}{q} \right \rfloor = \sum_{k = 1}^{q} \left \lfloor \frac{kp}{q} \right \rfloor\end{eqnarray}$$
17.
$$\sum_{n=1}^{+\infty}\frac{\mu(n)}{n}=1-\frac12-\frac13-\frac15+\frac16-\frac17+\frac1{10}-\frac1{11}-\frac1{13}+\frac1{14}+\frac1{15}-\cdots=0$$
18.
$$\frac{\pi}{4}=\sum_{n=1}^{\infty}\arctan\frac{1}{f_{2n+1}}$$
19.
$$\prod_{k=1}^{n-1}2\sin\frac{k \pi}{n} = n$$
20.
$$\frac{1 - \cos \alpha + \sin \alpha}{1 + \cos \alpha + \sin \alpha} = \sqrt{\frac{1-\cos \alpha}{1 + \cos \alpha}}$$
21.
$$\boxed{\boxed{\begin{align}E &=\sqrt{\left(pc\right)^{2} + \left(mc^{2}\right)^{2}}=mc^{2}+\left[\sqrt{\left(pc\right)^{2} \left(mc^{2}\right)^{2}} - mc^{2}\right]\\[3mm]&=mc^{2}+{\left(pc\right)^{2} \over \sqrt{\left(pc\right)^{2} \left(mc^{2}\right)^{2}} + mc^{2}}=mc^{2}+{p^{2}/2m \over 1 + {\sqrt{\left(pc\right)^{2} + \left(mc^{2}\right)^{2}} - mc^{2} \over 2mc^{2}}}\\[3mm]&=mc^{2}+{p^{2}/2m \over 1 + {p^{2}/2m \over \sqrt{\left(pc\right)^{2} + \left(mc^{2}\right)^{2}} + mc^{2}}}=mc^{2}+{p^{2}/2m \over 1 +{p^{2}/2m \over 1 + {p^{2}/2m \over \sqrt{\left(pc\right)^{2} + \left(mc^{2}\right)^{2}} - mc^{2}}}}\end{align}}}$$
22.
Euler sayısı $e$
Aurea Altın oran $\phi$
Euler-Mascheroni sabiti $\gamma$
ve $\pi$.
$$e\cdot\gamma\cdot\pi\cdot\phi \approx e + \gamma + \pi + \phi$$