Complex Arithmetic Formulas

\(\DeclareMathOperator{\sech}{sech}
\DeclareMathOperator{\csch}{csch}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\asin}{asin}
\DeclareMathOperator{\acos}{acos}
\DeclareMathOperator{\atan}{atan}
\DeclareMathOperator{\asec}{asec}
\DeclareMathOperator{\acot}{acot}
\DeclareMathOperator{\acsc}{acsc}
\DeclareMathOperator{\acosh}{acosh}
\DeclareMathOperator{\asinh}{asinh}
\DeclareMathOperator{\atanh}{atanh}
\DeclareMathOperator{\asech}{asech}
\DeclareMathOperator{\acsch}{acsch}
\DeclareMathOperator{\acoth}{acoth}
\newcommand{\parallelsum}{\mathbin{\!/\mkern-5mu/\!}}\)

This is a collection of complex arithmetic formulas written in \(\LaTeX\) and used in the HP-41 programs. It includes everything from power to trigonometric formulas.

$$
\begin{align}
z_1+z_2&=(\Re _1+\Re _2)+j \cdot (\Im _1+\Im _2) \\
z_1-z_2&=(\Re _1-\Re _2)+j \cdot (\Im _1-\Im _2) \\
z_1 \cdot z_2&=r_1 \cdot r_2 \cdot \mathrm{e}^{j \cdot (\Phi_1+\Phi_2)} \\
\frac{1}{z} &= \frac{1}{r} \cdot \mathrm{e}^{-j \cdot \Phi} \\
\frac{z_2}{z_1} &= \frac{r_1}{r_2} \cdot \mathrm{e}^{j \cdot (\Phi _1-\Phi _2)} \\
z_1 \parallelsum z_2 &= \frac{z_1 \cdot z_2}{z_1+z_2} \\
\mathrm{e}^{z} &=\mathrm{e}^{\Re} \cdot \sin (\Im) + j \cdot \mathrm{e}^{\Re} \cdot \cos (\Im) \\
\ln (z) &= \ln (r) + j \cdot \Phi \\
z_2^{z_1} &= r_1^{\Re _2} \cdot \mathrm{e}^{- \Im _2 \cdot \Phi _1} \cdot \mathrm{e}^{j \cdot (\Re _2 \cdot \Phi _1 + \Im _2 \cdot \ln (r_1))} \\
\sqrt[n]{z} &= r^{\frac{1}{n}} \cdot \mathrm{e}^{j \cdot \frac{ \Phi }{n}} \\
\sin (z) &= \sin (\Re ) \cdot \cosh (\Im ) + j \cdot \cos (\Re ) \cdot \sinh (\Im ) \\
\cos (z) &= \cos (\Re ) \cdot \cosh (\Im ) + j \cdot \sin (\Re ) \cdot \sinh (\Im ) \\
\tan (z) &= \frac{\sin(2 \cdot \Re)}{\cosh(2 \cdot \Im) + \cos(2 \cdot \Re)} + j \cdot \frac{\sinh(2 \cdot \Im)}{\cosh(2 \cdot \Im) + \cos (2 \cdot \Re)} \\
\csc (z) &= \frac{1}{\sin(z)} \\
\sec (z) &= \frac{1}{\cos(z)} \\
\cot (z) &= \frac{1}{\tan(z)} \\
\sinh (z) &= \cos(\Im) \cdot \sinh(\Re) + j \cdot \sin(\Im) \cdot \cosh(\Re) \\
\cosh (z) &= \cos(\Im) \cdot \cosh(\Re) + j \cdot \sin(\Im) \cdot \sinh(\Re) \\
\tanh (z) &= \frac{\sinh(2 \cdot \Im)}{\cosh(2 \cdot \Re)} + j \cdot \frac{\sin(2 \cdot \Im)}{\cosh(2 \cdot \Re) + \cos(2 \cdot\Im)} \\
\csch (z) &= \frac{1}{\sinh(z)} \\
\sech (z) &= \frac{1}{\cosh(z)} \\
\coth (z) &= \frac{1}{\tanh(z)} \\
\asin (z) &= \asin(b) +j \cdot \sgn(\Im ) \cdot \ln(a + \sqrt{a^{\mathrm{e}}}-1) &a\geq1 \land b \text{ in } [\mathrm{rad}]\\
\acos (z) &= \acos(b) +j \cdot \sgn(\Im ) \cdot \ln(a + \sqrt{a^{\mathrm{e}}}-1) &a\geq1 \land b \text{ in } [\mathrm{rad}]\\
\text{where} \nonumber \\
a &= \tfrac{1}{2} \left( \sqrt{(\Re +1)^{2} + \Im ^{2} } + \sqrt{ (\Re -1)^{2} + \Im^{2}} \right) \nonumber \\
b &= \tfrac{1}{2} \left( \sqrt{(\Re +1)^{2} + \Im ^{2} } – \sqrt{ (\Re -1)^{2} + \Im^{2}} \right) \nonumber \\
\sgn(g) &= \begin{cases}-1 & x < 0\\1 & x \geq 0\end{cases} \nonumber \\ \atan(z) &= \tfrac{1}{2} \cdot (\pi - \atan( \frac{1+ \Im}{\Re} ) -\atan ( \frac{1-\Im}{\Re} ) + j \cdot \tfrac{1}{4} \ln \left( \frac{(\frac{1+\Im}{\Re})^2 +1}{(\frac{1-\Im}{\Re})^2 +1} \right) \\ \acsc(z) &= \asin(\frac{1}{z}) \\ \asec(z) &= \asec(\frac{1}{z}) \\ \acot(z) &= \acot(\frac{1}{z}) \\ \asinh(z) &= -j \cdot \asin(j \cdot z) \\ \acosh(z) &= j \cdot \acos(z) \\ \atanh(z) &= -j \cdot \atan(j \cdot z) \\ \acsch(z) &= j \cdot \acsc(j \cdot z) \\ \asech(z) &= -j \cdot \asec(z) \\ \acoth(z) &= j \cdot \acot(j \cdot z) \\ \end{align} $$
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Passionately curious and stubbornly persistent. Enjoys to inspire and consult with others to exchange the poetry of logical ideas.

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