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Value of an expression involving complex numbers
- Thread starter Uniman
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chisigma
Well-known member
- Feb 13, 2012
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Re: Complex function
$\displaystyle 3^{3 + i\ \frac{8\ \pi}{\ln 3}}= e^{3\ \ln 3}= 27$
Kind regards
$\chi$ $\sigma$
You can use Euler's identity to 'discover' that is...
$\displaystyle 3^{3 + i\ \frac{8\ \pi}{\ln 3}}= e^{3\ \ln 3}= 27$
Kind regards
$\chi$ $\sigma$
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Re: Complex function
Hello, Uniman!
We have: .[tex]X \;=\;3^3\cdot3^{\frac{8\pi}{\ln(3)}i} \;=\;27\cdot 3^{\frac{8\pi}{\ln(3)}i} [/tex] .[1]
[tex]\text{Let }y \:=\:3^{\frac{8\pi}{\ln(3)}i} [/tex]
[tex]\text{Take logs: }\:\ln(y) \;=\;\ln\left(3^{\frac{8\pi}{\ln(3)}i}\right) \;=\;\frac{8\pi}{\ln(3)}i\cdot\ln(3) \;=\;8\pi i[/tex]
. . [tex]y \;=\;e^{8\pi i} \;=\;\left(e^{i\pi}\right)^8 \;=\;(\text{-}1)^8 \;=\;1[/tex]
Substitute into [1]: .[tex]X \;=\;27\cdot1 \;=\;\boxed{27}[/tex]
Hello, Uniman!
We have: .[tex]X \;=\;3^3\cdot3^{\frac{8\pi}{\ln(3)}i} \;=\;27\cdot 3^{\frac{8\pi}{\ln(3)}i} [/tex] .[1]
[tex]\text{Let }y \:=\:3^{\frac{8\pi}{\ln(3)}i} [/tex]
[tex]\text{Take logs: }\:\ln(y) \;=\;\ln\left(3^{\frac{8\pi}{\ln(3)}i}\right) \;=\;\frac{8\pi}{\ln(3)}i\cdot\ln(3) \;=\;8\pi i[/tex]
. . [tex]y \;=\;e^{8\pi i} \;=\;\left(e^{i\pi}\right)^8 \;=\;(\text{-}1)^8 \;=\;1[/tex]
Substitute into [1]: .[tex]X \;=\;27\cdot1 \;=\;\boxed{27}[/tex]