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# Complex Numbers

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Let z be a complex number such that |z| = 1. Find the largest possible value of |z^2 + 1|.

May 23, 2022

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The largest value would have to be 2.  I got that intuitively.

Then I set out to prove it.

Let

$$z=a+bi\\ |z|=\sqrt{a^2+b^2}=1\\ a^2+b^2=1\\$$

$$z^2+1 = a^2-b^2+2abi+1\\ z^2+1 = a^2-b^2+1+2abi\\ |z^2+1| =\sqrt{ (a^2-b^2+1)^2+(2ab)^2}\\ |z^2+1| =\sqrt{ (a^2+b^2-2b^2+1)^2+(2ab)^2}\\ |z^2+1| =\sqrt{ (1-2b^2+1)^2+(2ab)^2}\\ |z^2+1| =\sqrt{ (2-2b^2)^2+(2ab)^2}\\ |z^2+1| =\sqrt{ (4+4b^4-8b^2)+(2ab)^2}\\ |z^2+1| =2\sqrt{ (1+b^4-2b^2)+(ab)^2}\\ |z^2+1| =2\sqrt{ (1-b^2)^2+(ab)^2}\\ |z^2+1| =2\sqrt{ (a^2)^2+(ab)^2}\\ |z^2+1| =2a\sqrt{ (a^2)+(b)^2}\\ |z^2+1| =2a\sqrt{ 1}\\ |z^2+1| =2a\\ \text{But a must be between 0 and 1}\\ \text{So the largest possible value of } |z^2+1| \;\;is\;\;2$$

there is probably a much better way to do it.

May 25, 2022