1. $$\varphi=\dfrac{1+\sqrt{5}}{2}$$
2. $$\displaystyle\int_0^1 \dfrac{1}{(1+x^\varphi)^\varphi}dx=2^{\frac{1-\sqrt5}{2}}$$ pf) $$\begin{aligned} \dfrac{d}{dx}(1+x^\varphi)^{1-\varphi} & =\dfrac{\varphi(1-\varphi)x^{\varphi-1}}{(1+x^\varphi)^\varphi} \\ & =-\dfrac{x^{\varphi-1}}{(1+x^\varphi)^\varphi} \end{aligned}$$에서, $$\begin{aligned} & \displaystyle\int_0^1\dfrac{1}{(1+x^\varphi)^\varphi}dx \\ & =\displaystyle\int_0^1 \dfrac{1+x^\varphi-x^\varphi}{(1+x^\varphi)^\varphi}dx \\ & =\displaystyle\int_0^1 \left( \dfrac{1}{(1+x^\varphi)^{\varphi-1}}-\dfrac{x^\varphi}{(1+x^\varphi)^\varphi} \right)dx \\ & =\displaystyle\int_0^1 \left( x'(1+x^\varphi)^{1-\varphi}+x[(1+x^\varphi)^{1-\varphi}]' \right) dx \\ & =\left[ x(1+x^\varphi)^{1-\varphi} \right]_0^1 \\ & =2^{1-\varphi}=2^{\frac{1-\sqrt5}{2}}\end{aligned}$$이다. $\blacksquare$
3. $\varphi^2=\varphi+1$에서, $$\varphi=\sqrt{1+\varphi}=\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$$
$\varphi=\dfrac{1+\sqrt5}{2},\ \psi=\dfrac{1-\sqrt5}{2}$라 하자.
4. $$F_n=\dfrac{1}{\sqrt{5}} \left\{ \varphi^n-\psi^n \right\}$$
5. $$F_n=\dfrac{\varphi^n-\psi^n}{\varphi-\psi}$$
6. $$\displaystyle\lim_{n\to\infty}\dfrac{F_{n+1}}{F_n}=\varphi$$
7. $$\log\varphi=\dfrac{1}{2n-1}\displaystyle\int_0^{\frac{F_{2n}+F_{2n-2}}{2}}\dfrac{dx}{\sqrt{x^2+1}}$$
8. $$\log\varphi=\dfrac{1}{2n}\displaystyle\int_0^{\frac{F_{2n+1}+F_{2n-1}}{2}}\dfrac{dx}{\sqrt{x^2-1}}$$
9. $$\varphi=1+\displaystyle\sum_{n=2}^{\infty}\dfrac{(-1)^n}{F_nF_{n-1}}$$
10. $$\displaystyle\int_0^{\infty}\dfrac{\sqrt x}{x^2+2x+5}dx=\dfrac{\pi}{2\sqrt\varphi}$$
11. $$\varphi=2\cos(\dfrac{\pi}{5})=2\sin(\dfrac{3\pi}{10})$$
12. $$\displaystyle\int_0^{\infty} \ln \left( \dfrac{x^2-2kx+k^2}{x^2+2kx\cos \sqrt{\pi^2-\varphi}+k^2} \right) \dfrac{dx}{x}=\varphi$$
13. $$\displaystyle\int_0^{\infty} \dfrac{x^{\frac{\pi}{5}-1}}{1+x^{2\pi}}dx=\varphi$$
14. $$\dfrac{1}{(\sqrt{\varphi\sqrt 5})e^{\frac{2\pi}{5}}}=1+\dfrac{e^{-2\pi}}{1+\dfrac{e^{-4\pi}}{1+\frac{e^{-6\pi}}{1+\frac{e^{-8\pi}}{1+\cdots}}}}$$
Exponential 17 최정원
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