Chapter 17

In Problems 9-22, sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ 0 \leq \arg (z) \leq 2 \pi / 3 $$

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

Sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ |\arg (z)|<\pi / 4 $$

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

Find all values of \(z\) satisfying the given equation. \(\sinh z=-1\)

Verify the given result. $$ \frac{e^{z_{1}}}{e^{z_{2}}}=e^{z_{1}-z_{2}} $$

Show that the given function is not analytic at any point, but is differentiable along the indicated curve(s). \(f(z)=3 x^{2} y^{2}-6 x^{2} y^{2} i ;\) coordinate axes

In Problems 17-20, write the given number in the form \(a+i b\). $$ \frac{3-i}{2+3 i}+\frac{2-2 i}{1+5 i} $$

In Problems \(15-20\), find all values of \(z\) satisfying the given equation. $$ \sinh z=-1 $$

In Problems \(17-20\), verify the given result. $$ \frac{e^{z_{1}}}{e^{z_{2}}}=e^{z_{1}-z_{2}} $$

In Problems 15-18, evaluate the given function at the indicated points. \(f(z)=e^{x} \cos y+i e^{x} \sin y\) (a) \(\pi i / 4\) (b) \(-1-\pi i\) (c) \(3+\pi i / 3\)

In Problems 9-22, sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ |\arg (z)|<\pi / 4 $$

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

Sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ |z-i|>1 $$

Write each complex number in polar form. Then use either \((4)\) or \((5)\) to obtain a polar form of the given number. Write the polar form in the form \(a+i b\). $$ \frac{-i}{2-2 i} $$

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

Find all values of \(z\) satisfying the given equation. \(\cos z=\sin z\)

Verify the given result. $$ e^{z+\pi i}=e^{z-\pi i} $$

In Problems 19-22, the given limit exists. Find its value. $$ \lim _{z \rightarrow i}\left(4 z^{3}-5 z^{2}+4 z+1-5 i\right) $$

In Problems 17-20, write the given number in the form \(a+i b\). $$ \frac{(1-i)^{10}}{(1+i)^{3}} $$

In Problems \(15-20\), find all values of \(z\) satisfying the given equation. $$ \cos z=\sin z $$

In Problems 17-20, show that the given function is not analytic at any point, but is differentiable along the indicated curve(s). $$ f(z)=x^{3}+3 x y^{2}-x+i\left(y^{3}+3 x^{2} y-y\right) \text {; coordinate axes } $$

In Problems 9-22, sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ |z-i|>1 $$

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

Sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ |z-i|>0 $$

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

Find all values of \(z\) satisfying the given equation. \(\cos z=i \sin z\)

Verify the given result. $$ \left(e^{z}\right)^{n}=e^{n z}, n \text { an integer } $$

Show that the given function is not analytic at any point, but is differentiable along the indicated curve(s). \(f(z)=x^{2}-x+y+i\left(y^{2}-5 y-x\right) ; y=x+2\)

In Problems 17-20, write the given number in the form \(a+i b\). $$ 4 e^{\pi i / 3} e^{-\pi i / 4} $$

In Problems \(15-20\), find all values of \(z\) satisfying the given equation. $$ \cos z=i \sin z $$

In Problems \(17-20\), verify the given result. $$ \left(e^{z}\right)^{n}=e^{n z}, n \text { an integer } $$

In Problems 17-20, show that the given function is not analytic at any point, but is differentiable along the indicated curve(s). $$ f(z)=x^{2}-x+y+i\left(y^{2}-5 y-x\right) ; y=x+2 $$

In Problems 19-22, the given limit exists. Find its value. $$ \lim _{z \rightarrow 1-i} \frac{5 z-2 z+2}{z+1} $$

In Problems 9-22, sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ |z-i|>0 $$

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

Sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ 2<|z-i|<3 $$

Write the given number in the form \(a+i b\). $$ i(1-i)(2-i)(2+6 i) $$

Show that \(f(z)=e^{\bar{z}}\) is nowhere analytic.

The given limit exists. Find its value. $$ \lim _{z \rightarrow i} \frac{z^{4}-1}{z-i} $$

In Problems 21-24, sketch the set of points in the complex plane satisfying the given inequality. $$ \operatorname{Im}\left(z^{2}\right) \leq 2 $$

$$ \text { Show that } f(z)=e^{\bar{z}} \text { is nowhere analytic. } $$

In Problems 19-22, the given limit exists. Find its value. $$ \lim _{z \rightarrow i} z^{4}-1 $$

In Problems 9-22, sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ 2<|z-i|<3 $$

In Problems 1-26, write the given number in the form \(a+i b\). $$ i(1-i)(2-i)(2+6 i) $$

Sketch the set of points in the complex plane satisfying the given inequality. Determine whether the set is a domain. $$ 1 \leq|z-1-i|<2 $$

Use \((8)\) to compute the indicated power. $$ (2-2 i)^{5} $$

Write the given number in the form \(a+i b\). $$ (1+i)^{2}(1-i)^{3} $$

Use the definition of equality of complex numbers to find all values of \(z\) satisfying the given equation. \(\sin z=i \sinh 2\)

The given limit exists. Find its value. $$ \lim _{z \rightarrow 1+i} \frac{z^{2}-2 z+2}{z^{2}-2 i} $$