Summary of Complex Number Operations

Different forms

1.    $a+bi = r(\cos P+i\sin P) = re^{ip}$
where $r^2=a^2+b^2, \quad \tan P=\dfrac{b}{a}$,   or   $a=r\cos P, \quad b=r\sin P$

2.    $(a+bi)+(c+di) = (a+c)+(b+d)i$

3.    $(a+bi)-(c+di) = (a-c)+(b-d)i$

Multiplication

4.    $(a+bi)(c+di) = (ac-bd)+(ad+bc)i$

5.    $[r(\cos P+i\sin P)][s(\cos Q+i\sin Q)] = rs[\cos(P+Q)+i\sin(P+Q)]$

Division

6.    $\dfrac{a+bi}{c+di} = \dfrac{ac+bd}{c^2+d^2} + \dfrac{bc-ad}{c^2+d^2}i$

7.    $\dfrac{r(\cos P+i\sin P)}{s(\cos Q+i\sin Q)} = \dfrac{r}{s}[\cos(P-Q)+i\sin(P-Q)]$

Powers and Roots

8.    $[r(\cos P+i\sin P)]^n = r^n (\cos nP +i\sin nP)$,   for integer values of $n$.

9.    $[r(\cos P+i\sin P)]^{\frac{1}{n}} = r^{\frac{1}{n}} \left(\cos\dfrac{P+2k\pi}{n} +i\sin\dfrac{P+2k\pi}{n}\right)$,   for integer values of $k$ and $n$.

10.    $(a+bi)^{c+di} = r^c e^{-dP}\left[\cos(d\ln r+cP+2ck\pi)+i\sin(d\ln r+cP+2ck\pi)\right]$
where the complex number $a+bi$ has the trigonometric form $r(\cos P+i\sin P)$.

Logarithms

11.    $\ln {re^{iP}} = \ln r + iP + 2k\pi i$,   for integer values of $k$.

12.    $\log_{a+bi}(c+di) = \dfrac{\ln(a+bi)}{\ln(c+di)}$

Trigonometric functions

13.    $\cos x = \dfrac{e^{ix}+e^{-ix}}{2}$

14.    $\cos (a+bi) = \cos a \cosh b -i\sin a \sinh b$

15.    $\sin x = \dfrac{e^{ix}-e^{-ix}}{2i}$

16.    $\sin (a+bi) = \sin a \cosh b +i\cos a \sinh b$

Hyperbolic functions

17.    $\cosh x = \dfrac{e^{x}+e^{-x}}{2}$

18.    $\cosh (a+bi) = \cosh a \cos b +i\sinh a \sin b$

19.    $\sinh x = \dfrac{e^{x}-e^{-x}}{2}$

20.    $\sinh (a+bi) = \sinh a \cos b +i\cosh a \sin b$

Inverse Trigonometric Functions

21.    $\cos^{-1}x = -i\ln\left[ x \pm i\sqrt{1-x^2}\right]$

22.    $\sin^{-1}x = -i\ln\left[ ix \pm \sqrt{1-x^2}\right]$

23.    $\tan^{-1}x = \dfrac{i}{2} \ln\dfrac{i+x}{i-x}$

Inverse Hyperbolic Functions

24.    $\cosh^{-1}x = \ln\left[ x \pm \sqrt{x^2-1}\right]$

25.    $\sinh^{-1}x = \ln\left[ x \pm \sqrt{x^2+1}\right]$

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