Complex powers

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?}

So I decided to try and figure it out algebraically, here are the fruits of my labour.

Let znz_n be a complex number in the form an+bnia_n + b_ni where ana_n and bnb_n are real numbers.

z1=a1+b1i, z2=a2+b2iz_1 = a_1 + b_1i,~ z_2 = a_2 + b_2i

z1z2=(a1+b1i)(a2+b2i)\large z_1^{z_2} = (a_1 + b_1i)^{(a_2 + b_2i)}

zn=rneθniz_n = r_ne^{\theta_ni}

(r1eθ1i)(r2eθ2i)\large (r_1e^{\theta_1i})^{(r_2e^{\theta_2i})}

r1(r2eθ2i)(eθ1i)(r2eθ2i)\large r_1^{(r_2e^{\theta_2i})} \cdot (e^{\theta_1i})^{(r_2e^{\theta_2i})}

Let cis θ=cosθ+isinθcis~ \theta = \cos \theta + i\sin \theta

eθni=cis θne^{\theta_ni} = cis~ \theta_n

r1(r2cis θ2)(cis θ1)(r2cis θ2)\large r_1^{(r_2cis~ \theta_2)} \cdot (cis~ \theta_1)^{(r_2cis~ \theta_2)}

(cis θn)x=cis θnx(cis~ \theta_n)^x = cis~ \theta_nx

r1(r2cis θ2)cis (θ1(r2cis θ2))\large r_1^{(r_2cis~ \theta_2)} \cdot cis~ (\theta_1 \cdot (r_2cis~ \theta_2))

cis (θ1r2cis θ2)cis~ (\theta_1r_2cis~ \theta_2)

cis (θ1r2cosθ2+θ1r2isinθ2)cis~ (\theta_1r_2\cos \theta_2 + \theta_1r_2i\sin \theta_2)

cis (a+bi)=cis aebcis~ (a + bi) = \frac{cis~ a}{e^b}

Proof of the above statement here

cis (θ1r2cosθ2)e(θ1r2sinθ2)\large \frac{cis~ (\theta_1r_2\cos \theta_2)}{e^{(\theta_1r_2\sin \theta_2)}}

r1(r2cis θ2)cis (θ1r2cosθ2)e(θ1r2sinθ2)\large r_1^{(r_2cis~ \theta_2)} \cdot \frac{cis~ (\theta_1r_2\cos \theta_2)}{e^{(\theta_1r_2\sin \theta_2)}}

r1(r2cosθ2)r1(ir2sin θ2)cis (θ1r2cosθ2)e(θ1r2sinθ2)\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)}}

r1(r2cosθ2)(elnr1)(ir2sin θ2)cis (θ1r2cosθ2)e(θ1r2sinθ2)\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)}}

r1(r2cosθ2)eir2sinθ2lnr1cis (θ1r2cosθ2)e(θ1r2sinθ2)\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)}}

r1(r2cosθ2)cis (r2sinθ2lnr1)cis (θ1r2cosθ2)e(θ1r2sinθ2)\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)}}

exieyi=cis xcis ye^{xi} \cdot e^{yi} = cis~ x \cdot cis~ y

exieyi=exi+yi=ei(x+y)e^{xi} \cdot e^{yi} = e^{xi + yi} = e^{i(x + y)}

exieyi=cis (x+y)e^{xi} \cdot e^{yi} = cis~ (x + y)

cis xcis y=cis (x+y)cis~ x \cdot cis~ y = cis~ (x + y)

r1(r2cosθ2)cis (θ1r2cosθ2+r2sinθ2lnr1)e(θ1r2sinθ2)\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)}}

r1=a12+b12, r2=a22+b22r_1 = \sqrt{a_1^2 + b_1^2},~ r_2 = \sqrt{a_2^2 + b_2^2}

θ1=arctanb1a1, θ2=arctanb2a2\theta_1 = \arctan \frac{b_1}{a_1},~ \theta_2 = \arctan \frac{b_2}{a_2}

z1z2=a12+b12 (a22+b22cos (arctanb2a2))cis (a22+b22arctanb1a1cos(arctanb2a2)+a22+b22sin (arctanb2a2)lna12+b12)e(a22+b22arctanb1a1sin(arctanb2a2))\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)}}

Well that's a mouthful, so I tried to find a simpler version. What follows is my second attempt.

z1z2=(r1eθ1i)(a2+b2i)\large z_1^{z_2} = \left(r_1e^{\theta_1i}\right)^{(a_2 + b_2i)}

r1(a2+b2i)e(a2θ1ib2θ1)\large r_1^{(a_2 + b_2i)} \cdot e^{(a_2\theta_1i - b_2\theta_1)}

r1a2r1b2ieb2θ1ea2θ1i\large r_1^{a_2} \cdot r_1^{b_2i} \cdot e^{-b_2\theta_1} \cdot e^{a_2\theta_1i}

r1a2(elnr1)b2icis (a2θ1)eb2θ1\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}}

r1a2eib2lnr1cis (a2θ1)eb2θ1\large r_1^{a_2} \cdot e^{ib_2\ln r_1} \cdot \frac{cis~ (a_2\theta_1)}{e^{b_2\theta_1}}

r1a2cis (b2lnr1)cis (a2θ1)eb2θ1\large r_1^{a_2} \cdot cis~ (b_2\ln r_1) \cdot \frac{cis~ (a_2\theta_1)}{e^{b_2\theta_1}}

r1a2cis (a2θ1+b2lnr1)eb2θ1\large r_1^{a_2} \cdot \frac{cis~ (a_2\theta_1 + b_2\ln r_1)}{e^{b_2\theta_1}}

r1a2eb2θ1cis (a2θ1+b2lnr1)\large \frac{r_1^{a_2}}{e^{b_2\theta_1}} \cdot cis~ (a_2\theta_1 + b_2\ln r_1)

(elnr1)a2eb2θ1cis (a2θ1+b2lnr1)\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)

ea2lnr1eb2θ1cis (a2θ1+b2lnr1)\large \frac{e^{a_2\ln r_1}}{e^{b_2\theta_1}} \cdot cis~ (a_2\theta_1 + b_2\ln r_1)

ea2lnr1b2θ1cis (a2θ1+b2lnr1)\large e^{a_2\ln r_1 - b_2\theta_1} \cdot cis~ (a_2\theta_1 + b_2\ln r_1)

z1z2=ea2lna12+b12b2arctanb1a1cis (a2arctanb1a1+b2lna12+b12)\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)

Hope you enjoyed the note.

Note by Jack Rawlin
5 years, 5 months ago

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