Def :: Fm(x)=[m∈[0,3]∈Zc∈Z]=n→∞∑cx4·c+m(4c+m)!
ex=3∑m0Fm(x)=F0(x)+F1(x)+F2(x)+F3(x)
a−x=1axab+c=ab·aca1x = x √a
(a b) c = a b · c = (a c) b ≠ a b c ≡ a( b c )
ln A=ln1+x1−x=2·n→∞∑0mx2m+12m+1x=A−1A+1
\displaystyle{π=4·arctan\ 1=4\left[{\sum_{m\\ 0}^{n→0}\frac1{4m+1}-\sum_{m\\ 0}^{n→0}\frac1{4m+3}}\right]}
\begin{array}{rclrclrcl}F_0\left({-a}\right) & = & F_0\left({a}\right)\quad & F_0\left({i·b}\right) & = & 1·F_0\left({b}\right) & F_0\left({-i·b}\right)\quad & = & 1·F_0\left({b}\right)\\ F_1\left({-a}\right) & = & -F_1\left({a}\right) & F_1\left({i·b}\right) & = & i·F_1\left({b}\right) & F_1\left({-i·b}\right) & = & -i·F_1\left({b}\right)\\ F_2\left({-a}\right) & = & F_2\left({a}\right) & F_2\left({i·b}\right) & = & -1·F_2\left({b}\right) & F_2\left({-i·b}\right) & = & -1·F_2\left({b}\right)\\ F_3\left({-a}\right) & = & -F_3\left({a}\right) & F_3\left({i·b}\right) & = & -i·F_3\left({b}\right) & F_3\left({-i·b}\right) & = & i·F_3\left({b}\right) \end{array}
\begin{array}{rclrcccl}F_0\left({x}\right) & = & \displaystyle{\frac{ch\ x+cos\ x}2}\quad & ch\ x & = & F_0\left({x}\right)+F_2\left({x}\right) & = & \displaystyle{\frac{e^x+e^{-x}}2}\\ F_1\left({x}\right) & = & \displaystyle{\frac{sh\ x+sin\ x}2} & sh\ x & = & F_1\left({x}\right)+F_3\left({x}\right) & = & \displaystyle{\frac{e^x-e^{-x}}2}\\ F_2\left({x}\right) & = & \displaystyle{\frac{ch\ x-cos\ x}2} & cos\ x & = & F_0\left({x}\right)-F_2\left({x}\right) & = & \displaystyle{\frac{e^{i·x}+e^{-i·x}}2}\\ F_3\left({x}\right) & = & \displaystyle{\frac{sh\ x-sin\ x}2} & sin\ x & = & F_1\left({x}\right)-F_3\left({x}\right) & = & \displaystyle{\frac{e^{i·x}-e^{-i·x}}2} \end{array}
log_x\left({a·b}\right)=log_x\ a+log_x\ b
log_x\frac{a}b=log_x\ a-log_x\ b
log_x\ a^b=b·log_x\ a
\frac1x=x^{-1}\qquad \frac1{x^z}=x^{-z}
\displaystyle{a=x^{log_x\ a}}
log_x\ a=\frac{log_z\ a}{log_z\ x}=\frac1{log_a\ x}\quad \mathbf{NB!}
\boxed{ln\ w=ln\left|{w}\right|+i·arg\ w}
\displaystyle{\boxed{z^{\frac1{ln\ z}}=e^1=w^{\frac1{ln\ w}}}}
\displaystyle{\begin{align}i^{0±4·k} & = 1\\ i^{1±4·k} &= i\\ i^{2±4·k} & = -1\\ i^{3±4·k} & = -i\\ ±i & = ±\sqrt{-1}\\ ±ln\ 1 & = ±0\\ ±k·ln\ i & = ln\ i^{±k} = ±i·\frac{k·π}2\end{align}}
\begin{array}{rclrclrclrcl}ch\ ix & = & Cos\ x & ch\ x & = & Cos\ ix & ch\ -ix & = & Cos\ x & ch\ -ix & = & ch\ x \\ sh\ ix & = & i·Sin\ x & sh\ x & = & -i·Sin\ ix & sh\ -ix & = & -i·Sin\ x & sh\ -x & = & -sh\ x\\ Cos\ ix & = & ch\ x & Cos\ x & = & ch\ ix & Cos\ -ix & = & ch\ x & Cos\ -x & = & Cos\ x\\ Sin\ ix & = & i·sh\ x & Sin\ x & = & -i·sh\ ix & Sin\ -ix & = & -i·sh\ x & Sin\ -x & = & -Sin\ x \end{array}
See also : Complex Numbers
[Eop]
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