Chapter 2: Complex Numbers

Find the values of the indicated roots.

$\sqrt[\mathbf{3}]{\mathbf{2}\mathbf{i}\mathbf{-}\mathbf{2}}$

Express the following complex numbers in the x+iy form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others—try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

22.${\left(\frac{1-i}{\sqrt{2}}\right)}^{42}$

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

Find the values of the indicated roots:

$\sqrt[\mathbf{4}]{\mathbf{8}\mathbf{i}\sqrt{\mathbf{3}}\mathbf{-}\mathbf{8}}$

Find each of the following in rectangular $\mathit{a}\mathbf{+}\mathit{z}\mathit{b}$ form if $\mathit{z}\mathbf{=}\mathbf{2}\mathbf{-}{}_{\mathbf{t}}\mathbf{3}$; if $\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}{}_{\mathbf{z}}\mathit{y}$.

$\frac{\mathbf{1}\mathbf{+}\mathbf{z}}{\mathbf{1}\mathbf{-}\mathbf{z}}$.

Evaluate each of the following in x+ iyform, 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.

Express the following complex numbers in the x+iy form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others—try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

23. $\frac{{(1+i)}^{48}}{{(\sqrt{3}-i)}^{25}}$

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

cosh (x)

Verify the formulas in Problems 17 to 24.

$\mathbf{cosh}\mathbf{}\mathbf{iz}\mathbf{=}\mathbf{cosz}$

Find each of the following in rectangular $\mathit{a}\mathbf{+}{}_{\mathbf{z}}\mathit{b}$ form if $\mathit{z}\mathbf{=}\mathbf{2}\mathbf{-}{}_{\mathbf{t}}\mathbf{3}$ ; if $\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}{}_{\mathbf{z}}\mathit{y}$.

$\frac{\mathbf{z}}{\mathbf{z}}$.