# 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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