Is $e^{1\over 1-ix}$ the complex conjugate of $e^{1\over 1+ix}$? Is there a simple rule to compute complex conjugates without having to find $a+ib$ form? Thanks.
Complex conjugates?
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7"Is there a simple rule to compute complex conjugates?" - yes, replace all instances of $i$ with $-i$. For instance, the conjugate of $\exp(a-ib)$ is $\exp(a-(-i)b)=\exp(a+ib)$... – 2011-10-16
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0@J.M.: I think that ought to be an answer. – 2011-10-16
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0@J.M.isn'tamathematician Does your comment hold in any case, no matter whether your complex number is in the form $a+bi$, $e^{i\delta}$ or other? I calculated that the complex conjugate of $(a+bi)e^{i\delta}$ is indeed $(a-bi)e^{-i\delta}$. – 2017-06-26
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1@Karlo, it should. Proving it is so is a nice exercise. – 2017-07-26
1 Answers
Hint: $z$ and $w$ are complex conjugates if and only if $zw$ is a real number and $|z|=|w|$. You might wish to check both conditions for $z=\exp(1/u)$ and $w=\exp(1/\bar u)$ and any nonzero complex number $u$.
Edit To compute $|z|$ and $|w|$, you could start from the fact that, for every real numbers $a$ and $b$, $\exp(a+\mathrm ib)=\exp(a)\exp(\mathrm ib)$. Since $\exp(\mathrm ib)=\cos(b)+\mathrm i\sin(b)$ has absolute value $1$, $\exp(a+\mathrm ib)$ has absolute value $\exp(a)$. A second step could be to write $z=\exp(1/u)$ as $z=\exp(a+\mathrm ib)$, hence to write $1/u$ as $a+\mathrm ib$ for $u=1-\mathrm ix$, then to do the same for $1/\bar u=c+\mathrm id$, and finally to check that $a=c$ (hence $|z|=|w|$) and $b+d=0$ (hence $zw$ is a real number).
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0Thanks, Didier, it is clear that the 1st one is true. But I am not sure how to find the absolute value... – 2011-10-16
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0@Terri, in other words, $|z|=|w|$ for $z=\exp(t)$ and $w=\exp(s)$ if and only if $z/w=\exp(t-s)$ has absolute value $1$ if and only if $t-s$ has real part zero. (Of course, you should convince yourself of these before using them.) – 2011-10-16
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1@RossMillikan: you mean the absolute value of $a+bi$ is $\sqrt{a^2+b^2}$ (assuming $a$ and $b$ are real). – 2011-10-16
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1As Robert Israel said about $a+bi$. The absolute value of ez is exp(Re(z)) because the imaginary part of z just gives a rotation. – 2011-10-16