How to find all complex z with Im(z) = 1, 0 <= arg(z) <= pi/4 ?

[firechicken188]

How do you do this question?

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Sketch the region of the complex plane consisting of all complex numbers z which satisfy both of the following conditions:

. . . . .\(\displaystyle Im(z^2)\, =\, 1\)

. . . . .\(\displaystyle 0\, \leq\, \arg(z)\, \leq\, \dfrac{\pi}{4}\)

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I really don't get the first part Im(z^2) thingy

 

[Deleted member 4993]

How do you do this question?

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Sketch the region of the complex plane consisting of all complex numbers z which satisfy both of the following conditions:

. . . . .\(\displaystyle Im(z^2)\, =\, 1\)

. . . . .\(\displaystyle 0\, \leq\, \arg(z)\, \leq\, \dfrac{\pi}{4}\)

---

I really don't get the first part Im(z^2) thingy

z = x + iy

z² = ??

Im(z²) = ??

 

[pisrationalhahaha]

How do you do this question?

---

Sketch the region of the complex plane consisting of all complex numbers z which satisfy both of the following conditions:

. . . . .\(\displaystyle Im(z^2)\, =\, 1\)

. . . . .\(\displaystyle 0\, \leq\, \arg(z)\, \leq\, \dfrac{\pi}{4}\)

---

I really don't get the first part Im(z^2) thingy

\(\displaystyle z\, =\, x\, +\, iy\)

\(\displaystyle z^2\, =\, x^2\, -\, y^2\, +\, 2ixy\)

What is the imaginary part of the square? Set this equal to the number they gave you, and solve for "y=". What does the graph of this look like?

\(\displaystyle 0\, \leq\, \arg(z)\, \leq\, \dfrac{\pi}{4}\)

\(\displaystyle \arg(z^2)\, =\, 2\, \times\, arg(z)\)

\(\displaystyle \Rightarrow\, 0\, \leq\, \dfrac{\arg(z^2)}{2}\, \leq\, \dfrac{\pi}{4}\)

Solve for "arg(z^2)" in the middle. What does this tell you?

Please write back if you get stuck. Thank you!