Chapter 2: Complex Numbers

Using the fact that a complex equation is really two real equations, find the double angle formulas $(forsin2\theta ,cos2\theta )$by using equation 10.2.

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

sinh(4 + 3i) .

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

$\frac{\mathbf{2}\mathbf{+}\mathbf{3}\mathbf{i}}{\mathbf{1}\mathbf{-}\mathbf{i}}$.

Evaluate the following absolute square of a complex number (which arises in a problem in quantum mechanics). Assume a and b are real. Express your answer in terms of a hyperbolic function.

${\left|\frac{{\left(a+\mathrm{bi}\right)}^2{e}^b-{\left(a-\mathrm{bi}\right)}^2{e}^{-b}}{4{\mathrm{abie}}^{-\mathrm{ia}}}\right|}^{\mathbf{2}}$

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 the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

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

As in Problem 27, find the formulas for $\mathbf{(}\mathit{f}\mathit{o}\mathit{r}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{3}\mathit{\theta }\mathbf{,}\mathbf{}\mathbf{cos}\mathbf{3}\mathit{\theta }\mathbf{)}$.

Question. Express the following complex numbers in the 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.

24. $\frac{{\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{i}\sqrt{\mathbf{3}}\mathbf{)}}^{\mathbf{21}}}{{\mathbf{(}\mathbf{i}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{38}}}$

Question: Express the following complex numbers in the 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.

25.${\left[\frac{1\sqrt{2}}{1+i}\right]}^{12}$

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

${\left(1+2\mathit{i}\right)}^{\mathbf{3}}$.