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- Thread starter Leo Liu
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In summary, this is true:-Euler's Formula, ##e^{i\theta}= \cos{\theta} + i\sin{\theta}##, is the most important equation in mathematics-If z=reiθ, why does z¯=re−iθ?

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If you try to justify that formula, where do you get stuck?

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Leo Liu

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$$\bar z^2=r^2(\cos\theta-i\sin\theta)^2$$

$$\bar z^2=r^2(\cos(2\theta)-i\sin(2\theta))$$

$$\bar z^2=r^2(\cos2\theta+i\sin(-2\theta))$$

$$\bar z^2=r^2(\cos(-2\theta)+i\sin(-2\theta))$$

I got it. Thanks.

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$$\bar z^2 = (re^{-i\theta})^2 = r^2e^{-2i\theta} =r^2(\cos (-2\theta) +i\sin(-2\theta))$$

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Leo Liu

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I actually stated that I didn't want to use this method on the course chat haha.PeroK said:

$$\bar z^2 = (re^{-i\theta})^2 = r^2e^{-2i\theta} =r^2(\cos (-2\theta) +i\sin(-2\theta))$$

Thank you though! It is very neat.

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FactChecker

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You should study each part you doubt until it is intuitive to you. It is very fundamental and important.Leo Liu said:I actually stated that I didn't want to use this method on the course chat haha.

View attachment 293673

Thank you though! It is very neat.

In words explain why:

If ##z=re^{i\theta}##, why does ##\bar{z}=re^{-i\theta}##?

Then why does ##\bar{z}^2=re^{-i2\theta}##?

etc.

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Many mathematicians consider Euler's Formula, ##e^{i\theta} = \cos{\theta} + i\sin{\theta}##, to be the most important equation in mathematics. You should get very comfortable with using it.

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Leo Liu

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I guess it's because $$\text{cis}(-\theta)=e^{-i\theta}$$.FactChecker said:If z=reiθ, why does z¯=re−iθ?

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Yes. Negating the imaginary part of ##z=x+iy = r(\cos\theta+ i\sin\theta)## to get ##\bar{z}= x-iy = r(\cos\theta- i\sin\theta)## is the same as negating the argument, ##\theta##, to get ##\bar{z}=r(\cos{(-\theta)} + i\sin{(-\theta)})= r(\cos\theta- i\sin\theta)##.Leo Liu said:I guess it's because $$\text{cis}(-\theta)=e^{-i\theta}$$.

It looks like every line in your original post is correct (I didn't look hard at it.), but it is not clear how you got some lines and if you understood it. At least in the beginning, it is good practice to really spell everything out. You can skip some details after you are well beyond that level, but not before.

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The power of a complex conjugate is a mathematical operation that involves raising a complex number to a certain exponent. It is calculated by multiplying the complex number by itself a certain number of times, as determined by the exponent.

The power of a complex conjugate is different from the power of a real number because complex numbers have both a real and imaginary component, while real numbers only have a real component. This means that the power of a complex conjugate involves both multiplication and division, while the power of a real number only involves multiplication.

The complex conjugate is significant in the power operation because it allows us to simplify complex numbers and make calculations easier. By multiplying a complex number by its conjugate, we can eliminate the imaginary component and end up with a real number.

The power of a complex conjugate is related to the modulus of a complex number through the formula |z|^2 = z * z*, where z* is the complex conjugate of z. This means that the modulus of a complex number is equal to the complex number multiplied by its conjugate.

Yes, the power of a complex conjugate can be negative or fractional. This means that we can raise a complex number to a negative or fractional exponent, just like we can with real numbers. The resulting answer will still be a complex number.

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