Q4P - Mathematical Methods in Physical Sciences

Question

Express the following complex numbers in the$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$  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.

 

4.  ${{\mathit{e}}^{\left(1/3\right)}}^{\left(3+4\mathrm{\pi i}\right)}$

Step-by-Step Solution

Answer

The value of expression $e^{\left(1/3\right)\left(3+4\mathrm{\pi i}\right)}$  is$-1.36-2.35i$  and the graph is shown below.

Step 1: Given information

The Given complex number is$e^{\left(1/3\right)\left(3+4\mathrm{\pi i}\right)}$ .

Step 2: Definition of complex numbers

The numbers that are presented in the form of $\mathit{a}\mathbf{+}\mathit{i}\mathit{b}$, where,$\mathit{a}\mathbf{,}\mathit{b}$ are real numbers and$\mathbf{\text{'}}\mathit{i}\mathbf{\text{'}}$is an imaginary number called complex numbers.

Step 3: Find the value of r and θ

The exponential form of the complex number is$z=r{e}^{i\theta }$      … (1)

Let the given number be$z={e^{\left(1/3\right)}}^{\left(3+4\mathrm{\pi i}\right)}$.

Multiply the exponent as:

$z=e^{\left(1/3\right)\left(3+4\mathrm{\pi i}\right)}$

$z=e^{\left[1+\left(4\mathrm{\pi i}/3\right)\right]}$

$z=e.e^{\left(4\mathrm{\pi i}/3\right)}$             ....(2)

Compare the equations (1) and (2) and find the values of $r$ , and $\theta$ as:

$r=e$

$\theta =\left(\frac{4\mathrm{\pi }}{3}\right)$

Step 4: Use the standard and polar form of a complex number

The standard form of the complex number is $x+iy$  and the polar form is $r \cos \theta +ir \sin \theta$.

Compare both the above forms and put the values of $r$ , and $\theta$ as:

 

Solve for  $x$:

$x=r \cos \theta$

$x=e\cos \left(\frac{4\mathrm{\pi }}{3}\right)$ {from equation (1) and (2)}

 $x=-1.36$

Solve for  $y$:

 

$y=r \sin \theta$

$y=e\sin \left(\frac{4\mathrm{\pi }}{3}\right)$    {from equation (1) and (2)}

 $y=-2.35$

Put the values of  $x,y$ in the standard form of the complex number as:

$z=-1.36-2.35i$

Hence, the complex number is given by $z=-1.35-2.35i$ .

Step 5: Plot the complex number

The graph of $z=-1.36-2.35i$  as follows: