Basics :
\[\displaystyle{\boxed{\boxed{\boxed{\ i^{\ -1}\ =\ -i\ =\ \frac1i\ }}} \tag{ID:0}}\]
\[\displaystyle{\cases{z=\cases{a+ib\\ Re\ z+i·Im\ z\\ \left|{z}\right|·e^{\ i\ φ}\\ e^{\ ln\ \left|{z}\right|\ +\ i·arg\ z}}\\ {}\\ \mathbf{\overline{z}}=\cases{a-ib\\ Re\ z-i·Im\ z\\ \left|{z}\right|·e^{\ -i\ φ}\\ e^{\ ln\ \left|{z}\right|\ -\ i·arg\ z}}}}\]
\[\displaystyle{\cases{\left|{z}\right|=\left|{\sqrt{\ z·\mathbf{\overline{z}}\ \ {}}}\right|=\left|{\sqrt{Re^{\ 2}\ z-\left({i·Im\ z}\right)^2}}\right|=\left|{\sqrt{Re^{\ 2}\ z+Im^{\ 2}\ z}}\right| .\ .\ .\\ .\ .\ .\ {}_\text{ can be derived straight from } z {}_\text{ if exists such analytic } f\left({z}\right)=\mathbf{\overline{z}} {}_\text{ and } f\left({\mathbf{\overline{z}}}\right)=z}}\]
the above hints a possible flaw in the complex number definition ... also ...( some may find the following a miss definition : ) if we allow the arg z ∈ ℂ . . . the complex number won't hold/survive (( as any number can be any number in pure math . . . it is natural to suggest such))
\[\displaystyle{arg\ z=arctan\ \frac{Im\ z}{Re\ z}=arctan\ \frac{-i\ \left({\ z\ -\ \overline{z}\ {}}\right)}{z\ +\ \overline{z}}}\]
\[\displaystyle{z·c^{\ ±1}=e^{\ ln\ \left|{z}\right|\ ±\ ln\ \left|{c}\right|}·e^{\ i\ ·\ \left({arg\ z\ ±\ arg\ c}\right)}}\]
\[\displaystyle{z·c=\left({a+ib}\right)\left({x+iy}\right)=\left({ax-by}\right)+i·\left({bx+ay}\right)\tag{ID:1}}\]
\[\displaystyle{\frac zc=\frac{a+ib}{x+iy}=\frac{\left({a+ib}\right)\left({x-iy}\right)}{x^2-\left({iy}\right)^2}=\frac{ax+by+i\ \left({bx-ay}\right)}{x^2+y^2}=\frac{z·\overline{c}}{\left|{c}\right|^{\ 2}}\ ...\\ ...\ in\ case\ of\ \frac 1i=\frac{1·\left({-i}\right)}{\left|{i}\right|^2}\ also\ \frac 1{-i}=\frac{1·i}{\left|{-i}\right|^2}\ \tag{ID:0}}\]
\[\displaystyle{z^n=\left|{z}\right|^n\ ·\ e^{\ i\ ·\ n\ ·\ arg\ z}\ ,\ n\in\left[{\mathbb{Z^{-}},\mathbb{Z^{0}},\mathbb{Z^{+}}}\right]}\]
the above hints another ... "peculiarity" in the complex number's "nature" --e.g-- when the n → 0 (arc z → 0) the z gets flat near the [–1,1] "horizontal" (never the [ –i , i ] vertical --e.g.-- near "exponential zero" the ℂ → ℝ ) . . . PS! -- it does not necessarily pose a flaw in the complex number definition -- it just might hint the nature of Math detail
\[\displaystyle{z^{\frac1n}=\left|{z}\right|^{\frac1n}\ ·\ e^{\ i\ ·\ \frac{arg\ z\ ·\ 2kπ}n}\ ,\ k\in\left[{\ 0\ ,\ n-1\ }\right]\ ,\ n\in\mathbb{Z^{+}}}\]
\[\displaystyle{z^w=e^{w\ ln\ z}=e^{\ \left({Re\ w\ +\ i\ ·\ Im\ w\ }\right)\left({\ ln\ \left|{z}\right|\ +\ i\ ·\ arg\ z\ }\right)}=\\ =e^{\left({\ Re\ w\ ·\ ln\ \left|{z}\right|\ -\ Im\ w\ ·\ arg\ z\ }\right)}·e^{i\ ·\ \left({\ Im\ w\ ·\ ln\ \left|{z}\right|\ +\ Re\ w\ ·\ arg\ z\ }\right)}\tag{ID:1}}\]
See also : Exponential functions
[Eop]
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