LAC02 Complex Powers MRG 2012.0523 -- Sage

#1)R^R: 2^3=8 show((2+0*i)^(3+0*i))

 $\newcommand{\Bold}[1]{\mathbf{#1}}8$

#2)I^R: (2i)^2=-8i show((0+2*i)^(3+0*i))

 $\newcommand{\Bold}[1]{\mathbf{#1}}-8 i$

#3)C^R: (1+i)^3=-2+2i show((1+i)^(3+0*i)) print "(1+i)^0 = ",(1+i)^0 print "(1+i)^1 = ",(1+i)^1 print "(1+i)^2 = ",(1+i)^2 print "(1+i)^3 = ",(1+i)^3 show((1+x)^0) show((1+x)^1) show(expand((1+x)^2)) show(expand((1+x)^3))

 $\newcommand{\Bold}[1]{\mathbf{#1}}2 i - 2$ 
(1+i)^0 =  1
(1+i)^1 =  I + 1
(1+i)^2 =  2*I
(1+i)^3 =  2*I - 2
 $\newcommand{\Bold}[1]{\mathbf{#1}}1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}x + 1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} + 2 \, x + 1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}x^{3} + 3 \, x^{2} + 3 \, x + 1$

(1+i)^0 =  1
(1+i)^1 =  I + 1
(1+i)^2 =  2*I
(1+i)^3 =  2*I - 2

#4)R^I: 2^(3i)=??? show(((2+0*i)^(0+3*i)).n(digits=3)) show((2+0*i)^(0+3*i)) show((e^log(2))^(0+3*i)) show((e^(3*log(2)*i)).simplify_radical()) show((cos(3*log(2))+i*sin(3*log(2))).n(digits=3))

 $\newcommand{\Bold}[1]{\mathbf{#1}}-0.487 + 0.873i$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}2^{3 i}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}2^{3 i}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}2^{3 i}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-0.487 + 0.873i$

#5)I^I: (2sqrt(3)i)^(3i)=??? show(((0+2*sqrt(3)*i)^(0+3*i)).n(digits=3)) show(((0+2*sqrt(3)*i)^(0+3*i))) show(((0+2*sqrt(3))^(0+3*i)*(0+i)^(0+3*i))) show(((e^(log(2*sqrt(3))))^(0+3*i)*(e^(pi/2*i))^(0+3*i))) show((e^(3*log(2*sqrt(3))*i)*e^(3*pi/2*i^2)).n(digits=3)) show((e^(-3*pi/2)*(cos(3*log(2*sqrt(3)))+i*sin(3*log(2*sqrt(3))))).n(digits=3))

 $\newcommand{\Bold}[1]{\mathbf{#1}}-0.00749 - 0.00497i$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(2 i \, \sqrt{3}\right)^{3 i}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}i^{\left(3 i\right)} 2^{\left(3 i\right)} 3^{\left(\frac{3}{2} i\right)}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}i^{\left(3 i\right)} 2^{\left(3 i\right)} 3^{\left(\frac{3}{2} i\right)}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-0.00749 - 0.00497i$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-0.00749 - 0.00497i$

#6)C^I: (2+2sqrt(3))^(3i) show(((2+2*sqrt(3)*i)^(0+3*i)).n()) show(((2+2*sqrt(3)*i)^(0+3*i))) show(((4*(cos(pi/3)+i*sin(pi/3)))^(0+3*i))) show((4^(0+3*i)*(e^(pi/3*i))^(0+3*i)).n()) show((e^(-pi)*(cos(3*log(4))+i*sin(3*log(4)))).n())

 $\newcommand{\Bold}[1]{\mathbf{#1}}-0.0227163845992973 - 0.0367615097955021i$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}{\left(2 i \, \sqrt{3} + 2\right)}^{3 i}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}{\left(2 i \, \sqrt{3} + 2\right)}^{3 i}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-0.0227163845992973 - 0.0367615097955021i$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-0.0227163845992973 - 0.0367615097955021i$

#7)R^C: 2^(3+3i) show((2^(3+3*i)).n()) show((2^(3+3*i))) show((2^3*2^(3*i))) show((8*(cos(3*log(2))+i*sin(3*log(2)))).n())

 $\newcommand{\Bold}[1]{\mathbf{#1}}-3.89595534372625 + 6.98724065419897i$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}2^{3 i + 3}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}8 \, 2^{\left(3 i\right)}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-3.89595534372625 + 6.98724065419897i$

#8)I^C: (2sqrt(3)i)^(3+3i) show(((0+2*sqrt(3)*i)^(3+3*i)).n()) show(((0+2*sqrt(3)*i)^(3+3*i))) show((((2*sqrt(3))*e^(pi/2*i))^(3+3*i))) show(((2*sqrt(3))^3*(e^(pi/2*i))^3*(2*sqrt(3))^(3*i)*(e^(pi/2*i))^(3*i)).n()) show(((2*sqrt(3))^3*e^(-3*pi/2)*(cos(3*pi/2)+i*sin(3*pi/2))*(cos(3*log(2*sqrt(3)))+i*sin(3*log(2*sqrt(3))))).n())

 $\newcommand{\Bold}[1]{\mathbf{#1}}-0.206445759890775 + 0.311173447749976i$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(2 i \, \sqrt{3}\right)^{3 i + 3}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(2 i \, \sqrt{3}\right)^{3 i + 3}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-0.206445759890775 + 0.311173447749976i$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-0.206445759890775 + 0.311173447749976i$

#9)C^C: (2+2sqrt(3)i)^(3+3*i) show(((2+2*sqrt(3)*i)^(3+3*i)).n()) show(((2+2*sqrt(3)*i)^(3+3*i))) show(((4*e^(pi/3*i))^(3+3*i)).n()) show((4^3*(e^(pi/3*i))^3*4^(3*i)*(e^(pi/3*i))^(3*i)).n()) show((64*e^(-pi)*(cos(pi)+i*sin(pi))*(cos(3*log(4))+i*sin(3*log(4)))).n())

 $\newcommand{\Bold}[1]{\mathbf{#1}}1.45384861435503 + 2.35273662691213i$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}{\left(2 i \, \sqrt{3} + 2\right)}^{3 i + 3}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}1.45384861435503 + 2.35273662691213i$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}1.45384861435503 + 2.35273662691213i$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}1.45384861435503 + 2.35273662691213i$

LAC02 Complex Powers MRG 2012.0523

3670 days ago by lac2012