Chapter 17

In Problems 1-6, find the image of the given line under the mapping \(f(z)=z^{2}\) $$ y=2 $$

In Problems 1-8, sketch the graph of the given equation. $$ \operatorname{Re}(z)=5 $$

In Problems 1-26, write the given number in the form \(a+i b\). $$ 2 i^{3}-3 i^{2}+5 i $$

In Problems 1-14, find all values of the given quantity. \(\sin ^{-1}(-i)\)

In Problems 1-10, express \(e^{z}\) in the form \(a+i b\). \(z=\frac{\pi}{6} i\)

In Problems 1 and 2 , the given function is analytic for all \(z\). Show that the Cauchy-Riemann equations are satisfied at every point. $$ f(z)=z^{3} $$

Answer Problems 1-16 without referring back to the text. Fill in the blank or answer true/false. \(\operatorname{Re}(1+i)^{10}=\) _________ and \(\operatorname{Im}(1+i)^{10}=\) _________.

In Problems \(1-12\), express the given quantity in the form \(a+i b\). $$ \cos (3 i) $$

$$ \text { In Problems } 1-10 \text {, express } e^{z} \text { in the form } a+i b \text {. } $$ $$ z=\frac{\pi}{6} i $$

In Problems 1-10, write the given complex number in polar form. $$ 2 $$

Find the image of the given line under the mapping \(f(z)=z^{2}\) $$ x=-3 $$

Sketch the graph of the given equation. $$ \operatorname{Im}(z)=-2 $$

Write the given complex number in polar form. -10

Write the given number in the form \(a+i b\). $$ 3 i^{5}-i^{4}+7 i^{3}-10 i^{2}-9 $$

Express the given quantity in the form \(a+i b\). \(\sin (-2 i)\)

Express \(e^{z}\) in the form \(a+i b\). \(z=-\frac{\pi}{3} i\)

The given function is analytic for all \(z\). Show that the Cauchy-Riemann equations are satisfied at every point. $$ f(z)=3 z^{2}+5 z-6 i $$

Answer Problems 1-16 without referring back to the text. Fill in the blank or answer true/false. If \(z\) is a point in the third quadrant, then \(i z\) is in the _________ quadrant.

In Problems \(1-12\), express the given quantity in the form \(a+i b\). $$ \sin (-2 i) $$

$$ \text { In Problems } 1-10 \text {, express } e^{z} \text { in the form } a+i b \text {. } $$ $$ z=-\frac{\pi}{3} i $$

In Problems 1 and 2 , the given function is analytic for all \(z\). Show that the Cauchy-Riemann equations are satisfied at every point. $$ f(z)=3 z^{2}+5 z-6 i $$

In Problems 1-6, find the image of the given line under the mapping \(f(z)=z^{2}\). $$ x=-3 $$

In Problems 1-8, sketch the graph of the given equation. $$ \operatorname{Im}(z)=-2 $$

In Problems 1-10, write the given complex number in polar form. $$ -10 $$

In Problems 1-26, write the given number in the form \(a+i b\). $$ 3 i^{5}-i^{4}+7 i^{3}-10 i^{2}-9 $$

Find the image of the given line under the mapping \(f(z)=z^{2}\) $$ x=0 $$

Sketch the graph of the given equation. $$ \operatorname{Im}(\bar{z}+3 i)=6 $$

Write the given complex number in polar form. \(-3 i\)

Write the given number in the form \(a+i b\). $$ i^{8} $$

Find all values of the given quantity. \(\sin ^{-1} 0\)

Express the given quantity in the form \(a+i b\). \(\sin \left(\frac{\pi}{4}+i\right)\)

Express \(e^{z}\) in the form \(a+i b\). \(z=-1+\frac{\pi}{4} i\)

In Problems 3-8, show that the given function is not analytic at any point. $$ f(z)=\operatorname{Re}(z) $$

Answer Problems 1-16 without referring back to the text. Fill in the blank or answer true/false. $$ \text { If } z=3+4 i \text {, then } \operatorname{Re}\left(\frac{z}{\bar{z}}\right)= $$ _________.

In Problems 1-14, find all values of the given quantity. $$ \sin ^{-1} 0 $$

In Problems \(1-12\), express the given quantity in the form \(a+i b\). $$ \sin \left(\frac{\pi}{4}+i\right) $$

$$ \text { In Problems } 1-10 \text {, express } e^{z} \text { in the form } a+i b \text {. } $$ $$ z=-1+\frac{\pi}{4} i $$

In Problems 1-8, sketch the graph of the given equation. $$ \operatorname{Im}(\bar{z}+3 i)=6 $$

In Problems 1-10, write the given complex number in polar form. $$ -3 i $$

In Problems 1-26, write the given number in the form \(a+i b\). $$ i^{8} $$

Find the image of the given line under the mapping \(f(z)=z^{2}\) $$ y=0 $$

Sketch the graph of the given equation. $$ \operatorname{Im}(z-i)=\operatorname{Re}(z+4-3 i) $$

Write the given complex number in polar form. \(6 i\)

Write the given number in the form \(a+i b\). $$ i^{11} $$

Express the given quantity in the form \(a+i b\). \(\cos (2-4 i)\)

Express \(e^{z}\) in the form \(a+i b\). \(z=2-\frac{\pi}{2} i\)

Show that the given function is not analytic at any point. $$ f(z)=y+i x $$

Answer Problems 1-16 without referring back to the text. Fill in the blank or answer true/false. $$ i^{127}-5 i^{9}+2 i^{-1}= _________. $$

In Problems \(1-12\), express the given quantity in the form \(a+i b\). $$ \cos (2-4 i) $$

$$ \text { In Problems } 1-10 \text {, express } e^{z} \text { in the form } a+i b \text {. } $$ $$ z=2-\frac{\pi}{2} i $$