Chapter 17

Show that the given function is not analytic at any point. $$ f(x)=\frac{x}{x^{2}+y^{2}}+i \frac{y}{x^{2}+y^{2}} $$

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

$$ \text { In Problems } 1-10 \text {, express } e^{z} \text { in the form } a+i b \text {. } $$ $$ z=-0.3+0.5 i $$

In Problems 3-8, show that the given function is not analytic at any point. $$ f(x)=\frac{x}{x^{2}+y^{2}}+i \frac{y}{x^{2}+y^{2}} $$

In Problems 7-14, express the given function in the form \(f(z)=u+i v .\) $$ f(z)=7 z-9 i \bar{z}-3+2 i $$

In Problems 1-8, sketch the graph of the given equation. $$ |z+2+2 i|=2 $$

In Problems 1-10, write the given complex number in polar form. $$ -2-2 \sqrt{3} i $$

In Problems 1-26, write the given number in the form \(a+i b\). $$ i(4-i)+4 i(1+2 i) $$

Express the given function in the form \(f(z)=u+i v\) $$ f(z)=z^{2}-3 z+4 i $$

In Problems 9-22, sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ \operatorname{Re}(z)<-1 $$

Write the given complex number in polar form. \(\frac{3}{-1+i}\)

Write the given number in the form \(a+i b\). $$ (2-3 i)(4+i) $$

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

Express the given quantity in the form \(a+i b\). \(\cosh (\pi i)\)

Express \(e^{z}\) in the form \(a+i b\). \(z=5 i\)

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

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

$$ \text { In Problems } 1-10 \text {, express } e^{z} \text { in the form } a+i b \text {. } $$ $$ z=5 i $$

In Problems 7-14, express the given function in the form \(f(z)=u+i v .\) $$ f(z)=z^{2}-3 z+4 i $$

In Problems 1-10, write the given complex number in polar form. $$ \frac{3}{-1+i} $$

In Problems 1-26, write the given number in the form \(a+i b\). $$ (2-3 i)(4+i) $$

Sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ |\operatorname{Re}(z)|>2 $$

Write the given complex number in polar form. \(\frac{12}{\sqrt{3}+i}\)

Write the given number in the form \(a+i b\). $$ \left(\frac{1}{2}-\frac{1}{4} i\right)\left(\frac{2}{3}+\frac{5}{3} i\right) $$

Express the given quantity in the form \(a+i b\). \(\sinh \left(\frac{3 \pi}{2} i\right)\)

Express \(e^{z}\) in the form \(a+i b\). \(z=-0.23-i\)

Express the given function in the form \(f(z)=u+i v\) $$ f(z)=3 \bar{z}^{2}+2 z $$

Answer Problems 1-16 without referring back to the text. Fill in the blank or answer true/false. $$ \text { If }\left|e^{z}\right|=1 \text {, then } z \text { is a pure imaginary number. } $$ _________.

In Problems 1-14, find all values of the given quantity. $$ \tan ^{-1} 3 i $$

In Problems \(1-12\), express the given quantity in the form \(a+i b\). $$ \sinh \left(\frac{3 \pi}{2} i\right) $$

$$ \text { In Problems } 1-10 \text {, express } e^{z} \text { in the form } a+i b \text {. } $$ $$ z=-0.23-i $$

In Problems 7-14, express the given function in the form \(f(z)=u+i v .\) $$ f(z)=3 \bar{z}^{2}+2 z $$

In Problems 9-22, sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ |\operatorname{Re}(z)|>2 $$

In Problems 1-10, write the given complex number in polar form. $$ \frac{12}{\sqrt{3}+i} $$

In Problems 1-26, write the given number in the form \(a+i b\). $$ \left(\frac{1}{2}-\frac{1}{4} i\right)\left(\frac{2}{3}+\frac{5}{3} i\right) $$

Sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ \operatorname{Im}(z)>3 $$

In Problems 11-14, write the number given in polar form in the form \(a+i b\). $$ z=5\left(\cos \frac{7 \pi}{6}+i \sin \frac{7 \pi}{6}\right) $$

Write the given number in the form \(a+i b\). $$ (2+3 i)^{2} $$

Find all values of the given quantity. \(\sinh ^{-1} \frac{4}{3}\)

Express the given quantity in the form \(a+i b\). \(\sinh \left(1+\frac{\pi}{3} i\right)\)

Express the given function in the form \(f(z)=u+i v\) $$ f(z)=z^{3}-4 z $$

In Problems 1-14, find all values of the given quantity. $$ \sinh ^{-1} \frac{4}{3} $$

In Problems \(1-12\), express the given quantity in the form \(a+i b\). $$ \sinh \left(1+\frac{\pi}{3} i\right) $$

In Problems 7-14, express the given function in the form \(f(z)=u+i v .\) $$ f(z)=z^{3}-4 z $$

In Problems 9-22, sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ \operatorname{Im}(z)>3 $$

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

Sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ \operatorname{Im}(z-i)<5 $$

Write the number given in polar form in the form \(a+i b\). $$ z=8 \sqrt{2}\left(\cos \frac{11 \pi}{4}+i \sin \frac{11 \pi}{4}\right) $$

Write the given number in the form \(a+i b\). $$ (1-i)^{3} $$

Express the given quantity in the form \(a+i b\). \(\cosh (2+3 i)\)