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I have $(1+i)^{13}$ and I need to find the principal argument. I did this: $(1+i)^{13}$ = $(2)^{13/2}(cos(\frac{13pi}{4}) + isin(\frac{13pi}{4}))$ using De Moivre's Theorem, but I dont know where to go from here to find the principal arguement

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    Where did you encounter this problem? Most authors define a principal branch by excluding the negative real axis (from the complex plane), so that a principal "argument" would be an angle in $(-\pi/2,+\pi/2)$. However your textbook author (if there is one) may have a different convention. – hardmath Sep 16 '20 at 01:27
  • It was a past exam paper question at my uni –  Sep 16 '20 at 01:30
  • Note $(1+i)^2=2i, (1+i)^4=-4, (1+i)^8=16$ – J. W. Tanner Sep 16 '20 at 01:52

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$$(1+i)^{13}=(1+i)(((1+i)^2)^2)^3=(1+i)((2i)^2)^3=-64(1+i)=$$ $$=64\sqrt2\left(-\frac{1}{\sqrt2}-\frac{1}{\sqrt2}i\right)=64\sqrt2(\cos225^{\circ}+i\sin225^{\circ})=$$ $$=64\sqrt2(\cos(-135^{\circ})+i\sin(-135^{\circ})).$$

Michael Rozenberg
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As hardmath's comment indicated, it depends on the convention.
For example, one convention is that the principle angle must be in the range $(-\pi, \pi].$ This is the convention adopted by Palka in "Intro to Complex Function Theory." Another plausible convention is that the principle angle must be in the range $[0, 2\pi).$

I am going to arbitrarily adopt the Palka convention (above) in this answer. Note, that the problem can not be answered without first identifying which convention is to be used to identify the range for principle angles.

Your work identifies a pertinent angle as $(13\pi)/4.$ You have to identify the unique angle in $(-\pi, \pi]$ that is equivalent to $(13\pi)/4 \pmod{2\pi}.$

Since $(13\pi/4) + (-4\pi) = (-3\pi/4)$, the pertinent angle
(based on Palka's convention) is $(-3\pi/4)$.

user2661923
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You may be aware that multiplying complex numbers add up their arguments. This is a interactive demonstration of how multiplying two Complex numbers look like.

Now, your complex number is $z=1+i$, which looks like this on the complex plane:

enter image description here

Multiplying it by itself would double its angle. In other words, if you raise it to a power $n$ , the complex number would be turned $n$ times.

Also, it is evident that $8$ turns would bring it back to where it is already. So, we have the following diagram that illustrates the argument of $z^n$

enter image description here

Though I admit that this isn't a better method always, but it provides us an intuition of how to see the multiplication of complex numbers. It can add to your understanding and interpretation/manipulation capacity too. Cheers!

Soumyadwip Chanda
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