Complex Numbers

Verify each of the following by using equations (11.4), (12.2), and (12.3).

 $\mathbf{tanz}\mathbf{=}\mathbf{tan}\mathbf{(}\mathbf{ }\mathbf{x}\mathbf{+}\mathbf{iy}\mathbf{)}\mathbf{=}\frac{\mathbf{tanx}\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{tanhy}}{\mathbf{1}\mathbf{-}\mathbf{i}\mathbf{ }\mathbf{tanx}\mathbf{ }\mathbf{ }\mathbf{tanhy}}$

Verify each of the following by using equations (11.4), (12.2), and (12.3).

$\mathit{t}\mathit{a}\mathit{n}\mathit{h}\mathbf{ }\mathit{z}\mathbf{=}\frac{\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{h}\mathbf{ }\mathbf{x}\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{ }\mathbf{y}}{\mathbf{1}\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{h}\mathbf{ }\mathbf{x}\mathbf{ }\mathbf{ }\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{ }\mathbf{y}\mathbf{ }}$

Show that ${\mathbf{e}}^{\mathbf{nz}}\mathbf{=}\mathbf{(}\mathbf{coshz}\mathbf{+}\mathbf{sinhz}{\mathbf{)}}^{\mathbf{n}}\mathbf{=}\mathbf{cosh}\mathbf{ }\mathbf{nz}\mathbf{+}\mathbf{sinh}\mathbf{ }\mathbf{nz}\mathbf{.}$Use this and a similar equation for ${\mathit{e}}^{\mathbf{-}\mathbf{n}\mathbf{z}}$to find formulas for $\mathit{c}\mathit{o}\mathit{s}\mathit{h}\mathbf{3}\mathit{z}$and $\mathit{s}\mathit{i}\mathit{n}\mathit{h}\mathbf{3}\mathit{z}$in terms of $\mathbf{sinhz}$ and $\mathbf{coshz}$.

Use a computer to plot graphs of sinh(x), cosh(x), and tanh(x).

Using (12.2) and (8.1), find, in summation form, the power series for sinhx and coshx. Check the first few terms of your series by computer.

Find the real part, the imaginary part, and the absolute value of 

cosh (x)

Find the real part, the imaginary part, and the absolute value of 

 COS(ix)

Question: Find the real part, the imaginary part, and the absolute value of 

$\mathbf{sin}\mathbf{(}\mathit{x}\mathbf{-}\mathit{y}\mathbf{i}\mathbf{)}$ .

Question: Find the real part, the imaginary part, and the absolute value of

cosh(2 - 3i) .

Question: Find the real part, the imaginary part, and the absolute value of 

sinh(4 + 3i) .

Find the real part, the imaginary part, and the absolute value of

 $\mathit{t}\mathit{a}\mathit{n}\mathit{h}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{\pi i}\mathbf{)}$.

Find each of the following in the x+iy  form and check your answers by computer.

$\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left(2\pi i\right)$

Find each of the following in the x+iy  form and check your answers by computer. 

$\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\frac{3\pi i}{4}\right)$

Find each of the following in the x + iy form and check your answers by computer.

$\mathbf{sinh}\left(\ln 2 +\frac{i \pi }{3}\right)$

Find each of the following in the x + iy form and check your answers by computer.

$\mathbf{cosh}\left(\frac{\mathrm{i\pi }}{2}- \ln 3\right)$

Find each of the following in the x + iy form and check your answers by computer.

tani

Find each of the following in the x + iy form and check your answers by computer.

$\mathit{s}\mathit{i}\mathit{n}\frac{\mathbf{i}\mathbf{\pi }}{\mathbf{2}}$

Find each of the following in the x + iy form and check your answers by computer.

$\mathit{c}\mathit{o}\mathit{s}\mathbf{ }\mathit{h}\mathbf{(}\mathit{i}\mathit{\pi }\mathbf{+}\mathbf{2}\mathbf{)}$

Find each of the following in the x + iy form and check your answers by computer.

$\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(1+\frac{i\pi }{2}\right)$

Find each of the following in the x+iy form and check your answers by computer.

$\mathit{c}\mathbf{os}\mathbf{\left(}\mathbf{i}\mathbf{\pi }\mathbf{\right)}$

The functions $\mathit{s}\mathit{i}\mathit{n}\mathit{t}\mathbf{,}\mathbf{ }\mathit{c}\mathit{o}\mathit{s}\mathit{t}\mathbf{,}\mathbf{·}\mathbf{·}\mathbf{·}$, are called “circular functions” and the functions $\mathbf{sinh}\mathbf{ }\mathit{t}\mathbf{,}\mathbf{cosh}\mathbf{ }\mathit{t}\mathbf{,}\mathbf{·}\mathbf{·}\mathbf{·}$ , are called “hyperbolic functions”. To see a reason for this, show that $\mathit{x}\mathbf{=}\mathbf{cos}\mathbf{ }\mathit{t}\mathbf{,}\mathit{y}\mathbf{=}\mathbf{sin}\mathbf{ }\mathit{t}$ , satisfy the equation of a circle ${\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$ , while $\mathit{x}\mathbf{=}\mathbf{cosh}\mathbf{ }\mathit{t}\mathbf{,}\mathit{y}\mathbf{=}\mathbf{sinh}\mathbf{ }\mathit{t}$ , satisfy the equation of a hyperbola 

${\mathit{x}}^{\mathbf{2}}\mathbf{-}{\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$.

Evaluate each of the following in x+iy form, and compare with a computer solution.

$\mathit{l}\mathit{n}\mathbf{(}\mathbf{-}\mathit{e}\mathbf{)}$

Evaluate each of the following in $\mathit{x}\mathbf{ }\mathbf{+}\mathbf{ }\mathit{i}\mathit{y}$ form, and compare with a computer solution.

$\mathbf{ln}\mathbf{(}\mathbf{-}\mathit{i}\mathbf{)}$

Evaluate each of the following in $\mathit{x}\mathbf{ }\mathbf{+}\mathbf{ }\mathit{i}\mathit{y}$ form, and compare with a computer solution.

$\mathit{l}\mathit{n}\mathbf{(}\mathit{i}\mathbf{+}\sqrt{\mathbf{3}}\mathbf{)}$

Evaluate each of the following in x + iy  form, and compare with a computer solution.

ln ( i - 1 )

Evaluate each of the following in x+y form, and compare with a computer solution.

$\mathit{l}\mathit{n}\mathbf{(}\mathbf{-}\sqrt{\mathbf{2}}\mathbf{ }\mathit{i}\mathbf{-}\sqrt{\mathbf{2}}\mathbf{)}$

Question: Evaluate each of the following in x+y  form, and compare with a computer solution.\

$\ln \left(\frac{1-i}{\sqrt{2}}\right)$

Evaluate each of the following in x+iy form, and compare with a computer solution.

$\mathit{l}\mathit{n}\mathbf{\left(}\frac{\mathbf{ }\mathbf{1}\mathbf{+}\mathbf{i}}{\mathbf{1}\mathbf{-}\mathbf{i}}\mathbf{\right)}$

Evaluate each of the following in $\mathbf{x}\mathbf{+}\mathbf{iy}$ form, and compare with a computer solution ${\mathbf{i}}^{\mathbf{2}\mathbf{/}\mathbf{3}}$.

Evaluate each of the following in x+iy  form, and compare with a computer solution ${\left(-1\right)}^{\mathbf{i}}$.

Evaluate each of the following in  $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form, and compare with a computer solution.

$\mathbf{\left(}\mathit{i}\mathbf{\right)}\mathit{l}{\mathit{n}}^{\mathbf{i}}$

Evaluate each of the following in x+iy form, and compare with a computer solution ${\mathbf{2}}^{\mathbf{i}}$.

Evaluate each of the following in $\mathbf{x}\mathbf{+}\mathbf{iy}$ form, and compare with a computer solution.

${\mathbf{\left(}\mathbf{i}\mathbf{\right)}}^{\mathbf{3}\mathbf{+}\mathbf{i}}$

Evaluate each of the following in x + iy form, and compare with a computer solution.

${\mathit{i}}^{\mathit{2}\mathit{i}\mathit{/}\mathit{\pi }}$

Evaluate each of the following in $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form, and compare with a computer solution.${\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{i}}$

Evaluate each of the following in x + iyform, and compare with a computer solution.

${\left(\frac{1+i\sqrt{3}}{2}\right)}^{\mathbf{\text{'}}}$

Evaluate each of the following in x + iyform, and compare with a computer solution.

${\left(i-1\right)}^{\mathbf{l}\mathbf{+}\mathbf{1}}$

Evaluate each of the following in x + iyform, and compare with a computer solution.

$\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathbf{2}\mathit{i}\mathbf{ }\mathit{I}\mathit{n}\mathbf{ }\mathit{i}\mathbf{)}$

Evaluate each of the following in x + iy form, and compare with a computer solution.

 $\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathbf{\pi }\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{I}\mathbf{n}\left(2\right)\mathbf{)}$

Evaluate each of the following in x + iy form, and compare with a computer solution.

$\mathbf{sin}\left(i \ln \frac{1-i}{1+i}\right)$

Evaluate each of the following in$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form, and compare with a computer solution.$\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathbf{i}\mathbf{l}\mathbf{n}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{\right)}$

Evaluate each of the following in x + iy form, and compare with a computer solution.

$\mathbf{sin}\mathbf{(}\mathbf{i}\mathbf{ }\mathbf{ln}\left(\frac{\sqrt{3}+i}{2}\right)\mathbf{)}$

Evaluate each of the following in x + iy form, and compare with a computer solution.$\mathbf{(}\mathbf{1}\mathbf{-}\sqrt{\mathbf{2}\mathbf{i}}{\mathbf{)}}^{\mathbf{i}}$Hint: Find $\sqrt{\mathbf{2}\mathbf{i}}$ first.

Show that ${\mathbf{\left(}{\mathbf{a}}^{\mathbf{b}}\mathbf{\right)}}^{\mathbf{c}}$  can have more values than.${{\mathit{a}}^{\mathbf{b}}}^{\mathbf{c}}$ As examples compare 

1. ${\mathbf{\left[}{\mathbf{(}\mathbf{-}\mathbf{i}\mathbf{)}}^{\mathbf{2}\mathbf{+}\mathbf{i}}\mathbf{\right]}}^{\mathbf{2}\mathbf{-}\mathbf{i}}\mathbf{\text{ and }}{{\mathbf{(}\mathbf{-}\mathbf{i}\mathbf{)}}^{\mathbf{(}\mathbf{2}\mathbf{+}\mathbf{i}\mathbf{)}}}^{\mathbf{(}\mathbf{2}\mathbf{-}\mathbf{i}\mathbf{)}}\mathbf{=}{\mathbf{(}\mathbf{-}\mathbf{i}\mathbf{)}}^{\mathbf{5}}$ ;
2. ${\mathbf{\left(}{\mathbf{i}}^{\mathbf{i}}\mathbf{\right)}}^{\mathbf{i}}\mathbf{\text{ and }}{\mathbf{(}\mathbf{-}\mathbf{i}\mathbf{)}}^{\mathbf{-}\mathbf{1}}$ .

Find each of the following in the x + iyform and compare a computer solution$\mathit{a}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathbf{2}\mathbf{\right)}$.

Find each of the following in the x + iy form and compare a computer solution.

arctan2i

Question: Find each of the following in the x+iy form and compare a computer solution.

${\mathbf{sinh}}^{\mathbf{-}\mathbf{1}}\mathbf{(}\mathbf{i}\mathbf{/}\mathbf{2}\mathbf{)}$

Question: Find each of the following in the x + iy form and compare a computer solution.

$\mathbf{arccos}\left(i\sqrt{8}\right)$

Question: Find each of the following in the x + iy form and compare a computer solution.

${\mathbf{tanh}}^{\mathbf{-}\mathbf{1}}\mathbf{(}\mathbf{-}\mathit{i}\mathbf{)}$

Question: Find each of the following in the x + iy form and compare a computer solution.

$\mathbf{arctan}\mathbf{\left(}\mathit{i}\sqrt{\mathbf{2}}\mathbf{\right)}$