\[\displaystyle{Def\ ::\ F_m\left({x}\right)=\left[{\begin{array}{c}m\in\left[{0,3}\right]\in\mathbb{Z}\\ c\in\mathbb{Z}\end{array}}\right]=\sum_c^{n→∞}\frac{x^{4·c+m}}{\left({4c+m}\right)!}} \]
\[\displaystyle{e^x=\sum_{m\\ 0}^3F_m\left({x}\right)=F_0\left({x}\right)+F_1\left({x}\right)+F_2\left({x}\right)+F_3\left({x}\right)}\]
\[\displaystyle{\boxed{a^{-x}=\frac1{a^x}}\qquad \boxed{a^{b+c}=a^b·a^c}\qquad \boxed{a^{\frac1x}\ =\ \sqrt[x\ \ \ ]{a}}}\]
\[\displaystyle{\boxed{\boxed{\left({a^{\ b}}\right)^{\ c}\ =\ a^{\ b\ ·\ c}\ =\ \left({a^{\ c}}\right)^{\ b} \ \ne \ a^{\ b^{\ c}} \ \equiv \ a^{\left({\ b^{\ c}\ }\right)}}}}\]
\[\displaystyle{ln\ A=ln\frac{1+x}{1-x}=2·\sum_{0\\ m}^{n→∞}\frac{x^{2m+1}}{2m+1}\\ x=\frac{A-1}{A+1}}\]
\[\displaystyle{π=4·arctan\ 1=4\left[{\sum_{m\\ 0}^{n→0}\frac1{4m+1}-\sum_{m\\ 0}^{n→0}\frac1{4m+3}}\right]}\]
\[\begin{array}{rclrclrcl}F_0\left({-a}\right) & = & F_0\left({a}\right)\quad & F_0\left({i·b}\right) & = & 1·F_0\left({b}\right) & F_0\left({-i·b}\right)\quad & = & 1·F_0\left({b}\right)\\
F_1\left({-a}\right) & = & -F_1\left({a}\right) & F_1\left({i·b}\right) & = & i·F_1\left({b}\right) & F_1\left({-i·b}\right) & = & -i·F_1\left({b}\right)\\
F_2\left({-a}\right) & = & F_2\left({a}\right) & F_2\left({i·b}\right) & = & -1·F_2\left({b}\right) & F_2\left({-i·b}\right) & = & -1·F_2\left({b}\right)\\
F_3\left({-a}\right) & = & -F_3\left({a}\right) & F_3\left({i·b}\right) & = & -i·F_3\left({b}\right) & F_3\left({-i·b}\right) & = & i·F_3\left({b}\right)
\end{array}\]
\[\begin{array}{rclrcccl}F_0\left({x}\right) & = & \displaystyle{\frac{ch\ x+cos\ x}2}\quad & ch\ x & = & F_0\left({x}\right)+F_2\left({x}\right) & = & \displaystyle{\frac{e^x+e^{-x}}2}\\
F_1\left({x}\right) & = & \displaystyle{\frac{sh\ x+sin\ x}2}
& sh\ x & = & F_1\left({x}\right)+F_3\left({x}\right) & =
& \displaystyle{\frac{e^x-e^{-x}}2}\\
F_2\left({x}\right) & = & \displaystyle{\frac{ch\ x-cos\ x}2}
& cos\ x & = & F_0\left({x}\right)-F_2\left({x}\right) & =
& \displaystyle{\frac{e^{i·x}+e^{-i·x}}2}\\
F_3\left({x}\right) & = & \displaystyle{\frac{sh\ x-sin\ x}2}
& sin\ x & = & F_1\left({x}\right)-F_3\left({x}\right) & =
& \displaystyle{\frac{e^{i·x}-e^{-i·x}}2}
\end{array}\]
\[log_x\left({a·b}\right)=log_x\ a+log_x\ b\]
\[log_x\frac{a}b=log_x\ a-log_x\ b\]
\[log_x\ a^b=b·log_x\ a\]
\[\frac1x=x^{-1}\qquad \frac1{x^z}=x^{-z}\]
\[\displaystyle{a=x^{log_x\ a}}\]
\[log_x\ a=\frac{log_z\ a}{log_z\ x}=\frac1{log_a\ x}\quad \mathbf{NB!}\]
\[\boxed{ln\ w=ln\left|{w}\right|+i·arg\ w}\]
\[\displaystyle{\boxed{z^{\frac1{ln\ z}}=e^1=w^{\frac1{ln\ w}}}}\]
\[\displaystyle{\begin{align}i^{0±4·k} & = 1\\ i^{1±4·k} &= i\\ i^{2±4·k} & = -1\\ i^{3±4·k} & = -i\\ ±i & = ±\sqrt{-1}\\ ±ln\ 1 & = ±0\\ ±k·ln\ i & = ln\ i^{±k} = ±i·\frac{k·π}2\end{align}}\]
\[\begin{array}{rclrclrclrcl}ch\ ix & = & Cos\ x & ch\ x & = & Cos\ ix & ch\ -ix & = & Cos\ x & ch\ -ix & = & ch\ x \\ sh\ ix & = & i·Sin\ x & sh\ x & = & -i·Sin\ ix & sh\ -ix & = & -i·Sin\ x & sh\ -x & = & -sh\ x\\ Cos\ ix & = & ch\ x & Cos\ x & = & ch\ ix & Cos\ -ix & = & ch\ x & Cos\ -x & = & Cos\ x\\ Sin\ ix & = & i·sh\ x & Sin\ x & = & -i·sh\ ix & Sin\ -ix & = & -i·sh\ x & Sin\ -x & = & -Sin\ x \end{array}\]
See also : Complex Numbers
[Eop]
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