- #1
AxiomOfChoice
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What's the complex conjugate of
[tex]
\frac{1}{\sqrt{1+it}}, \quad t \geq 0.
[/tex]
[tex]
\frac{1}{\sqrt{1+it}}, \quad t \geq 0.
[/tex]
The key is that for all complex z, [itex]z \overline{z} = |z|^2[/itex] so that [itex]\overline{z} = \frac{|z|^2}{z}[/itex]AxiomOfChoice said:What's the complex conjugate of
[tex]
\frac{1}{\sqrt{1+it}}, \quad t \geq 0.
[/tex]
It works out OK in this case :-) I did actually think about that. If you picture the operations on the complex plane, it's pretty easy to see.Hurkyl said:(Don't forget about branch cuts! A little bit of care must be used to ensure that the function and its proposed conjugate make consistent choices of principal value)
A complex conjugate is a mathematical operation that involves changing the sign of the imaginary component of a complex number. It is represented by adding a bar or asterisk above the number. For example, the complex conjugate of 3+4i would be 3-4i.
Taking the complex conjugate is useful in many mathematical applications, such as solving equations, finding roots, and simplifying expressions. It also helps with visualizing complex numbers on a graph, as the conjugate of a complex number represents a reflection across the real axis.
The complex conjugate of a complex number is found by changing the sign of the imaginary component of the number. For a complex number written in the form a+bi, the complex conjugate is a-bi. In other words, the real part stays the same and the imaginary part becomes its negative.
The complex conjugate of a complex number differs from the original number only in the sign of its imaginary component. This means that the real parts of both numbers are the same, but the imaginary parts are opposite. For example, the complex conjugate of 3+4i is 3-4i.
Yes, the complex conjugate of a real number is the number itself. This is because a real number has an imaginary component of zero, so changing the sign of the imaginary component will not have any effect. Therefore, the complex conjugate of a real number a is simply a.