A thought popped into my head the other day:
What would you get when you put a complex number to the power of another complex number? \text{What would you get when you put a complex number to the power of another complex number?} What would you get when you put a complex number to the power of another complex number?
So I decided to try and figure it out algebraically, here are the fruits of my labour.
Let z n z_n z n a n + b n i a_n + b_ni a n + b n i a n a_n a n b n b_n b n
z 1 = a 1 + b 1 i , z 2 = a 2 + b 2 i z_1 = a_1 + b_1i,~ z_2 = a_2 + b_2i z 1 = a 1 + b 1 i , z 2 = a 2 + b 2 i
z 1 z 2 = ( a 1 + b 1 i ) ( a 2 + b 2 i ) \large z_1^{z_2} = (a_1 + b_1i)^{(a_2 + b_2i)} z 1 z 2 = ( a 1 + b 1 i ) ( a 2 + b 2 i )
z n = r n e θ n i z_n = r_ne^{\theta_ni} z n = r n e θ n i
( r 1 e θ 1 i ) ( r 2 e θ 2 i ) \large (r_1e^{\theta_1i})^{(r_2e^{\theta_2i})} ( r 1 e θ 1 i ) ( r 2 e θ 2 i )
r 1 ( r 2 e θ 2 i ) ⋅ ( e θ 1 i ) ( r 2 e θ 2 i ) \large r_1^{(r_2e^{\theta_2i})} \cdot (e^{\theta_1i})^{(r_2e^{\theta_2i})} r 1 ( r 2 e θ 2 i ) ⋅ ( e θ 1 i ) ( r 2 e θ 2 i )
Let c i s θ = cos θ + i sin θ cis~ \theta = \cos \theta + i\sin \theta c i s θ = cos θ + i sin θ
e θ n i = c i s θ n e^{\theta_ni} = cis~ \theta_n e θ n i = c i s θ n
r 1 ( r 2 c i s θ 2 ) ⋅ ( c i s θ 1 ) ( r 2 c i s θ 2 ) \large r_1^{(r_2cis~ \theta_2)} \cdot (cis~ \theta_1)^{(r_2cis~ \theta_2)} r 1 ( r 2 c i s θ 2 ) ⋅ ( c i s θ 1 ) ( r 2 c i s θ 2 )
( c i s θ n ) x = c i s θ n x (cis~ \theta_n)^x = cis~ \theta_nx ( c i s θ n ) x = c i s θ n x
r 1 ( r 2 c i s θ 2 ) ⋅ c i s ( θ 1 ⋅ ( r 2 c i s θ 2 ) ) \large r_1^{(r_2cis~ \theta_2)} \cdot cis~ (\theta_1 \cdot (r_2cis~ \theta_2)) r 1 ( r 2 c i s θ 2 ) ⋅ c i s ( θ 1 ⋅ ( r 2 c i s θ 2 ) )
c i s ( θ 1 r 2 c i s θ 2 ) cis~ (\theta_1r_2cis~ \theta_2) c i s ( θ 1 r 2 c i s θ 2 )
c i s ( θ 1 r 2 cos θ 2 + θ 1 r 2 i sin θ 2 ) cis~ (\theta_1r_2\cos \theta_2 + \theta_1r_2i\sin \theta_2) c i s ( θ 1 r 2 cos θ 2 + θ 1 r 2 i sin θ 2 )
c i s ( a + b i ) = c i s a e b cis~ (a + bi) = \frac{cis~ a}{e^b} c i s ( a + b i ) = e b c i s a
Proof of the above statement here
c i s ( θ 1 r 2 cos θ 2 ) e ( θ 1 r 2 sin θ 2 ) \large \frac{cis~ (\theta_1r_2\cos \theta_2)}{e^{(\theta_1r_2\sin \theta_2)}} e ( θ 1 r 2 sin θ 2 ) c i s ( θ 1 r 2 cos θ 2 )
r 1 ( r 2 c i s θ 2 ) ⋅ c i s ( θ 1 r 2 cos θ 2 ) e ( θ 1 r 2 sin θ 2 ) \large r_1^{(r_2cis~ \theta_2)} \cdot \frac{cis~ (\theta_1r_2\cos \theta_2)}{e^{(\theta_1r_2\sin \theta_2)}} r 1 ( r 2 c i s θ 2 ) ⋅ e ( θ 1 r 2 sin θ 2 ) c i s ( θ 1 r 2 cos θ 2 )
r 1 ( r 2 cos θ 2 ) ⋅ r 1 ( i r 2 s i n θ 2 ) ⋅ c i s ( θ 1 r 2 cos θ 2 ) e ( θ 1 r 2 sin θ 2 ) \large r_1^{(r_2\cos \theta_2)} \cdot r_1^{(ir_2sin~ \theta_2)} \cdot \frac{cis~ (\theta_1r_2\cos \theta_2)}{e^{(\theta_1r_2\sin \theta_2)}} r 1 ( r 2 cos θ 2 ) ⋅ r 1 ( i r 2 s i n θ 2 ) ⋅ e ( θ 1 r 2 sin θ 2 ) c i s ( θ 1 r 2 cos θ 2 )
r 1 ( r 2 cos θ 2 ) ⋅ ( e ln r 1 ) ( i r 2 s i n θ 2 ) ⋅ c i s ( θ 1 r 2 cos θ 2 ) e ( θ 1 r 2 sin θ 2 ) \large r_1^{(r_2\cos \theta_2)} \cdot \left(e^{\ln r_1}\right)^{(ir_2sin~ \theta_2)} \cdot \frac{cis~ (\theta_1r_2\cos \theta_2)}{e^{(\theta_1r_2\sin \theta_2)}} r 1 ( r 2 cos θ 2 ) ⋅ ( e ln r 1 ) ( i r 2 s i n θ 2 ) ⋅ e ( θ 1 r 2 sin θ 2 ) c i s ( θ 1 r 2 cos θ 2 )
r 1 ( r 2 cos θ 2 ) ⋅ e i r 2 sin θ 2 ln r 1 ⋅ c i s ( θ 1 r 2 cos θ 2 ) e ( θ 1 r 2 sin θ 2 ) \large r_1^{(r_2\cos \theta_2)} \cdot e^{ir_2\sin \theta_2 \ln r_1} \cdot \frac{cis~ (\theta_1r_2\cos \theta_2)}{e^{(\theta_1r_2\sin \theta_2)}} r 1 ( r 2 cos θ 2 ) ⋅ e i r 2 sin θ 2 ln r 1 ⋅ e ( θ 1 r 2 sin θ 2 ) c i s ( θ 1 r 2 cos θ 2 )
r 1 ( r 2 cos θ 2 ) ⋅ c i s ( r 2 sin θ 2 ln r 1 ) ⋅ c i s ( θ 1 r 2 cos θ 2 ) e ( θ 1 r 2 sin θ 2 ) \large r_1^{(r_2\cos \theta_2)} \cdot cis~ (r_2\sin \theta_2 \ln r_1) \cdot \frac{cis~ (\theta_1r_2\cos \theta_2)}{e^{(\theta_1r_2\sin \theta_2)}} r 1 ( r 2 cos θ 2 ) ⋅ c i s ( r 2 sin θ 2 ln r 1 ) ⋅ e ( θ 1 r 2 sin θ 2 ) c i s ( θ 1 r 2 cos θ 2 )
e x i ⋅ e y i = c i s x ⋅ c i s y e^{xi} \cdot e^{yi} = cis~ x \cdot cis~ y e x i ⋅ e y i = c i s x ⋅ c i s y
e x i ⋅ e y i = e x i + y i = e i ( x + y ) e^{xi} \cdot e^{yi} = e^{xi + yi} = e^{i(x + y)} e x i ⋅ e y i = e x i + y i = e i ( x + y )
e x i ⋅ e y i = c i s ( x + y ) e^{xi} \cdot e^{yi} = cis~ (x + y) e x i ⋅ e y i = c i s ( x + y )
c i s x ⋅ c i s y = c i s ( x + y ) cis~ x \cdot cis~ y = cis~ (x + y) c i s x ⋅ c i s y = c i s ( x + y )
r 1 ( r 2 cos θ 2 ) ⋅ c i s ( θ 1 r 2 cos θ 2 + r 2 sin θ 2 ln r 1 ) e ( θ 1 r 2 sin θ 2 ) \large r_1^{(r_2\cos \theta_2)} \cdot \frac{cis~ (\theta_1r_2\cos \theta_2 + r_2\sin \theta_2 \ln r_1)}{e^{(\theta_1r_2\sin \theta_2)}} r 1 ( r 2 cos θ 2 ) ⋅ e ( θ 1 r 2 sin θ 2 ) c i s ( θ 1 r 2 cos θ 2 + r 2 sin θ 2 ln r 1 )
r 1 = a 1 2 + b 1 2 , r 2 = a 2 2 + b 2 2 r_1 = \sqrt{a_1^2 + b_1^2},~ r_2 = \sqrt{a_2^2 + b_2^2} r 1 = a 1 2 + b 1 2 , r 2 = a 2 2 + b 2 2
θ 1 = arctan b 1 a 1 , θ 2 = arctan b 2 a 2 \theta_1 = \arctan \frac{b_1}{a_1},~ \theta_2 = \arctan \frac{b_2}{a_2} θ 1 = arctan a 1 b 1 , θ 2 = arctan a 2 b 2
z 1 z 2 = a 1 2 + b 1 2 ( a 2 2 + b 2 2 cos ( arctan b 2 a 2 ) ) ⋅ c i s ( a 2 2 + b 2 2 arctan b 1 a 1 cos ( arctan b 2 a 2 ) + a 2 2 + b 2 2 sin ( arctan b 2 a 2 ) ln a 1 2 + b 1 2 ) e ( a 2 2 + b 2 2 arctan b 1 a 1 sin ( arctan b 2 a 2 ) ) \large z_1^{z_2} = \sqrt{a_1^2 + b_1^2}^{~\left(\sqrt{a_2^2 + b_2^2}\cos~ \left(\arctan \frac{b_2}{a_2}\right)\right)} \cdot \frac{cis~ \left(\sqrt{a_2^2 + b_2^2}\arctan \frac{b_1}{a_1} \cos \left(\arctan \frac{b_2}{a_2}\right) + \sqrt{a_2^2 + b_2^2}\sin~ \left(\arctan \frac{b_2}{a_2}\right)\ln \sqrt{a_1^2 + b_1^2}\right)}{e^{\left(\sqrt{a_2^2 + b_2^2} \arctan \frac{b_1}{a_1} \sin \left(\arctan \frac{b_2}{a_2}\right)\right)}} z 1 z 2 = a 1 2 + b 1 2 ( a 2 2 + b 2 2 cos ( arctan a 2 b 2 ) ) ⋅ e ( a 2 2 + b 2 2 arctan a 1 b 1 sin ( arctan a 2 b 2 ) ) c i s ( a 2 2 + b 2 2 arctan a 1 b 1 cos ( arctan a 2 b 2 ) + a 2 2 + b 2 2 sin ( arctan a 2 b 2 ) ln a 1 2 + b 1 2 )
Well that's a mouthful, so I tried to find a simpler version. What follows is my second attempt.
z 1 z 2 = ( r 1 e θ 1 i ) ( a 2 + b 2 i ) \large z_1^{z_2} = \left(r_1e^{\theta_1i}\right)^{(a_2 + b_2i)} z 1 z 2 = ( r 1 e θ 1 i ) ( a 2 + b 2 i )
r 1 ( a 2 + b 2 i ) ⋅ e ( a 2 θ 1 i − b 2 θ 1 ) \large r_1^{(a_2 + b_2i)} \cdot e^{(a_2\theta_1i - b_2\theta_1)} r 1 ( a 2 + b 2 i ) ⋅ e ( a 2 θ 1 i − b 2 θ 1 )
r 1 a 2 ⋅ r 1 b 2 i ⋅ e − b 2 θ 1 ⋅ e a 2 θ 1 i \large r_1^{a_2} \cdot r_1^{b_2i} \cdot e^{-b_2\theta_1} \cdot e^{a_2\theta_1i} r 1 a 2 ⋅ r 1 b 2 i ⋅ e − b 2 θ 1 ⋅ e a 2 θ 1 i
r 1 a 2 ⋅ ( e ln r 1 ) b 2 i ⋅ c i s ( a 2 θ 1 ) e b 2 θ 1 \large r_1^{a_2} \cdot \left(e^{\ln r_1}\right)^{b_2i} \cdot \frac{cis~ (a_2\theta_1)}{e^{b_2\theta_1}} r 1 a 2 ⋅ ( e ln r 1 ) b 2 i ⋅ e b 2 θ 1 c i s ( a 2 θ 1 )
r 1 a 2 ⋅ e i b 2 ln r 1 ⋅ c i s ( a 2 θ 1 ) e b 2 θ 1 \large r_1^{a_2} \cdot e^{ib_2\ln r_1} \cdot \frac{cis~ (a_2\theta_1)}{e^{b_2\theta_1}} r 1 a 2 ⋅ e i b 2 ln r 1 ⋅ e b 2 θ 1 c i s ( a 2 θ 1 )
r 1 a 2 ⋅ c i s ( b 2 ln r 1 ) ⋅ c i s ( a 2 θ 1 ) e b 2 θ 1 \large r_1^{a_2} \cdot cis~ (b_2\ln r_1) \cdot \frac{cis~ (a_2\theta_1)}{e^{b_2\theta_1}} r 1 a 2 ⋅ c i s ( b 2 ln r 1 ) ⋅ e b 2 θ 1 c i s ( a 2 θ 1 )
r 1 a 2 ⋅ c i s ( a 2 θ 1 + b 2 ln r 1 ) e b 2 θ 1 \large r_1^{a_2} \cdot \frac{cis~ (a_2\theta_1 + b_2\ln r_1)}{e^{b_2\theta_1}} r 1 a 2 ⋅ e b 2 θ 1 c i s ( a 2 θ 1 + b 2 ln r 1 )
r 1 a 2 e b 2 θ 1 ⋅ c i s ( a 2 θ 1 + b 2 ln r 1 ) \large \frac{r_1^{a_2}}{e^{b_2\theta_1}} \cdot cis~ (a_2\theta_1 + b_2\ln r_1) e b 2 θ 1 r 1 a 2 ⋅ c i s ( a 2 θ 1 + b 2 ln r 1 )
( e ln r 1 ) a 2 e b 2 θ 1 ⋅ c i s ( a 2 θ 1 + b 2 ln r 1 ) \large \frac{\left(e^{\ln r_1}\right)^{a_2}}{e^{b_2\theta_1}} \cdot cis~ (a_2\theta_1 + b_2\ln r_1) e b 2 θ 1 ( e ln r 1 ) a 2 ⋅ c i s ( a 2 θ 1 + b 2 ln r 1 )
e a 2 ln r 1 e b 2 θ 1 ⋅ c i s ( a 2 θ 1 + b 2 ln r 1 ) \large \frac{e^{a_2\ln r_1}}{e^{b_2\theta_1}} \cdot cis~ (a_2\theta_1 + b_2\ln r_1) e b 2 θ 1 e a 2 ln r 1 ⋅ c i s ( a 2 θ 1 + b 2 ln r 1 )
e a 2 ln r 1 − b 2 θ 1 ⋅ c i s ( a 2 θ 1 + b 2 ln r 1 ) \large e^{a_2\ln r_1 - b_2\theta_1} \cdot cis~ (a_2\theta_1 + b_2\ln r_1) e a 2 ln r 1 − b 2 θ 1 ⋅ c i s ( a 2 θ 1 + b 2 ln r 1 )
z 1 z 2 = e a 2 ln a 1 2 + b 1 2 − b 2 arctan b 1 a 1 ⋅ c i s ( a 2 arctan b 1 a 1 + b 2 ln a 1 2 + b 1 2 ) \large z_1^{z_2} = e^{a_2\ln \sqrt{a_1^2 + b_1^2} - b_2\arctan \frac{b_1}{a_1}} \cdot cis~ \left(a_2\arctan \frac{b_1}{a_1} + b_2\ln \sqrt{a_1^2 + b_1^2}\right) z 1 z 2 = e a 2 ln a 1 2 + b 1 2 − b 2 arctan a 1 b 1 ⋅ c i s ( a 2 arctan a 1 b 1 + b 2 ln a 1 2 + b 1 2 )
Hope you enjoyed the note.
#Algebra
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