## Announce

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## Derivative

\lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \lim_{\delta x \to 0} \frac{f(a + \delta x) - f(a)}{\delta x}

## Power and Logarithm

$$ax^2 + bx + c = 0 \to x = \frac{-b + \sqrt{b^2 -4ac}}{2a}$$ $$a^{2+3} = a^2 a^3$$ $$a^{2 3} = (a^2)^3$$

a^{\log_a x} = x

log_a (xy) = log_a x + log_a y

\log_a x^p = p \ \log_a x

\log_a {1 \over x} = \log_a x^{-1} = -\log_a x

\log_a {x \over y} = \log_a \left(x \cdot {1 \over y}\right) = \log_a x -\log_a y

\left(\log_a x\right) \left(\log_b a\right) = \log_b a^{\log_a x} = \log_b x

より

\log_a x = {\log_b x \over \log_b a}

となる。これを底の変換という。正の実数 x が 1 でないならば、b = x とすることにより

\log_a x = {1 \over \log_x a}

\log_{1 \over a} x = {\log_a x \over \log_a \frac{1}{a}} = - \log_a x

## Series and Integral

a + ar + ... + ar^{n-1} = \sum^{n-1}_{i=0} ar^i = \frac{1 - r^n}{1 - r}
a + ar + ... + ar^{n-1} + ...= \sum^{\infty}_{i=0} ar^i = \frac{1}{1 - r}\quad\mbox{ if |r| < 1}
\sum^{\infty}_{i=0} i r^i = {r \over (1 - r)^2} \quad\mbox{ for } |r| < 1
S_n = 1r^0 + 2r^1 + 3r^2 + .... nr^{n-1}
S_n - r S_n = 1r^0 + 1r^1 + 1r^2 + .... 1r^{n-1}

テーラー展開?

## e

:$e^{i x} = \cos x + i \;\sin x$ :$e^{-i x} = \cos x - i \;\sin x$ :$e^x = \cosh x + \sinh x\!$ :$e^{-x} = \cosh x - \sinh x.\!$ :$\frac{e^{i x} + e^{-i x}}{2} = \cos x$ :$\frac{e^{i x} - e^{-i x}}{2} = i \sin x$ :$\sinh x = \frac{e^x - e^{-x}}{2} = -i \sin ix \!$ :$\cosh x = \frac{e^{x} + e^{-x}}{2} = \cos ix \!$

{1 \over 2 \pi} \int_{- \pi}^{\pi} e^{j \omega n} d\omega = \begin{cases} \delta(n) = 1, & n = 0 \\ 0, & \text{otherwise} \end{cases}

because .... clear for n = 0, and derive for n != 0.

\int_{0}^{\infty} e^{-\lambda x} dx = 1/\lambda = ([\frac{1}{-\lambda}e^{-\lambda x}]_0^\infty)
e = \lim_{x \to 0} (1 + x)^{1/x} = \lim_{x \to \infty} (1 + 1/x)^x
• x = a の周りでの Taylor expansion
\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^{n}
\sum_{i=0}^{\infty} \frac{\lambda^i e^{-\lambda}}{i!} s^i = e^{\lambda(s-1)}
• 例：e の　x = 0 の周りでの Taylor expansion (マクローリン展開)
e^{x} = \sum^{\infty}_{n=0} \frac{x^n}{n!}\quad\mbox{ for all }x
e = \sum^{\infty}_{n=0} \frac{1}{n!}
ln(1+x) = \sum^{\infty}_{n=0} \frac{(-1)^{n+1}}{n!} x^n
\int_{-\infty}^{\infty} e^{-x^2/2} dx = \sqrt{2 \pi}

## Lagrange Multipler

Maximize $f(\textbf{x})$ subject to $g(\textbf{x}) <= c$

Let $L(\textbf{x}, \lambda) = f(\textbf{x}) + \lambda(g(\textbf{x}) -c )$

Take derivative respect to $x_i$ and $\lambda$ and set 0.