Math2.org Math Tables:
Complexity
(
Math
)
Basic Operations
Complex Definitions of Functions and Operations
(sum of squares)
(DeMoivre's Theorem)
if
then there are n complex nth roots (z) of w for k=0,1,..n-1
\[z(k) = r
1/n
[ \cos \frac{\theta + 2 \pi k}{n} + i \sin \frac{\theta + 2 \pi k}{n} ]\]
![\[i = \sqrt{-1}\]](/math/tex/804f6c73efb4d4e488e92fa31968c5d9.png "\[i = \sqrt{-1}\]")
![\[i^2 = -1\]](/math/tex/53318a21aacf998adff68e61d156701c.png "\[i^2 = -1\]")
![\[1 / i = -i\]](/math/tex/ab886d56a0c52ea044fea95bd60fb081.png "\[1 / i = -i\]")
![\[i<sup>4k</sup> = 1; i<sup>4k+1</sup> = i; i<sup>4k+2</sup> = -1; i<sup>4k+3</sup> = -i
\, \text{(integer k)}\]](/math/tex/a50c8f94f5d397cdcf8f9e1f240eec82.png "\[i<sup>4k</sup> = 1; i<sup>4k+1</sup> = i; i<sup>4k+2</sup> = -1; i<sup>4k+3</sup> = -i
\, \text{(integer k)}\]")
![\[\sqrt{i} = \sqrt{1/2} + \sqrt{1/2} i\]](/math/tex/a2316529ccb57ec6eeabb8e3f46bbf25.png "\[\sqrt{i} = \sqrt{1/2} + \sqrt{1/2} i\]")
![\[(a + bi) + (c + di) = (a+c) + (b + d) i\]](/math/tex/e3051ef70a88e843f10788b162d8faef.png "\[(a + bi) + (c + di) = (a+c) + (b + d) i\]")
![\[(a + bi) (c + di) = ac + adi + bci + bdi<sup>2</sup> = (ac - bd) + (ad +bc) i\]](/math/tex/5be3a47431361703c8fc32e1b7feb6f5.png "\[(a + bi) (c + di) = ac + adi + bci + bdi<sup>2</sup> = (ac - bd) + (ad +bc) i\]")
![\[\frac{1}{a + bi} = \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2} i\]](/math/tex/9ca4586e0acf7925d70356a3d22dc36c.png "\[\frac{1}{a + bi} = \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2} i\]")
![\[\frac{a + bi}{c + di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2} i\]](/math/tex/97a1c022b417ce99c3e997ddaa5a1d59.png "\[\frac{a + bi}{c + di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2} i\]")
![\[a<sup>2</sup> + b<sup>2</sup> = (a + bi) (a - bi)\]](/math/tex/9271b47b9f3e9f4cafc2dc9f8e85da04.png "\[a<sup>2</sup> + b<sup>2</sup> = (a + bi) (a - bi)\]")
(sum of squares)
![\[e<sup>i \theta</sup> = \cos(\theta) + i \sin(\theta)\]](/math/tex/1996ec2098e4b06f62c93fcb72caf484.png "\[e<sup>i \theta</sup> = \cos(\theta) + i \sin(\theta)\]")
![\[n<sup>a + bi</sup> = n^a (\cos(b \ln n) + i \sin(b \ln n)) \]](/math/tex/0eb3a4391dc8127686ef0137629a3954.png "\[n<sup>a + bi</sup> = n^a (\cos(b \ln n) + i \sin(b \ln n)) \]")
![\[\text{If} \, z = r(\cos \theta + i \sin \theta) \,
\text{then} \, z<sup>n</sup> = r<sup>n</sup> ( \cos n \theta + i \sin n \theta ) \]](/math/tex/8e671db515825a69b7ca860d3646da16.png "\[\text{If} \, z = r(\cos \theta + i \sin \theta) \,
\text{then} \, z<sup>n</sup> = r<sup>n</sup> ( \cos n \theta + i \sin n \theta ) \]")
(DeMoivre's Theorem)
![\[w = r(\cos \theta + i \sin \theta);n=integer\]](/math/tex/9a40fb5b0a0e9345c9985df094e531ef.png "\[w = r(\cos \theta + i \sin \theta);n=integer\]")
then there are n complex nth roots (z) of w for k=0,1,..n-1
![\[\text{If} \, z = r (\cos \theta + i \sin \theta) \,
\text{then} \, ln(z) = \ln r + i \theta\]](/math/tex/3901aeed28fb4538b7745e1e46fd96d4.png "\[\text{If} \, z = r (\cos \theta + i \sin \theta) \,
\text{then} \, ln(z) = \ln r + i \theta\]")
![\[sin(a + bi) = sin(a)cosh(b) + cos(a)sinh(b) i\]](/math/tex/fe7e28fc78121d48cc2bba54c56160ca.png "\[sin(a + bi) = sin(a)cosh(b) + cos(a)sinh(b) i\]")
![\[cos(a + bi) = cos(a)cosh(b) - sin(a)sinh(b) i\]](/math/tex/624491fcc576c0f8ad93927f57257e98.png "\[cos(a + bi) = cos(a)cosh(b) - sin(a)sinh(b) i\]")
![\[tan(a + bi) = \frac{\tan(a) + i \tanh(b)}{1 - i \tan(a) \tanh(b)} \]](/math/tex/1ca897416fb544d97b698f0456a4798a.png "\[tan(a + bi) = \frac{\tan(a) + i \tanh(b)}{1 - i \tan(a) \tanh(b)} \]")
![\[ = \frac{sech^2(b)\tan(a) + \sec^2(a)tanh(b) i}{1 + \tan^2(a)tanh^2(b)}\]](/math/tex/6ca1a9ed1cc53bf7caff852f03fbf9b6.png "\[ = \frac{sech^2(b)\tan(a) + \sec^2(a)tanh(b) i}{1 + \tan^2(a)tanh^2(b)}\]")
