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Chapter 2: Q18P (page 51)

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find x,y,r,θin your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

3ei2x.

Short Answer

Expert verified

The required values are mentioned below:

x=0,y=3,r=3,θ=π2

The graph of the number and its conjugate is shown below:

Step by step solution

01

Given Information 

The complex number is 3ei2x.

02

Definition of the Complex number.

A complex number is a combination of real and imaginary numbers.z=a+bz Where a and b are real numbers and z is the complex number.

03

Find the value

The formula is mentioned below:

x+iy=rcosθ+isinθ=reiθ

3ei2x=3cosπ2+isinπ2

The formulas for x and y are given below:

x=rcosθy=rsinθ

Find x and y:

x=3cosπ2=0y=3sinπ2=3

Find the value of θ.

Compare with the value of x:

x=3cosπ2r=3

Find the value of θ.

Compare with the value of x:

x=3cosπ2θ=π2

The value of the number becomes as follows:

3cosπ2+isinπ2=3eix/2

The graph of the complex number (blue) and its conjugate (red) is shown below:

The required values are mentioned below:

x=0,y=3.r=3,θ=π2

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Most popular questions from this chapter

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.

17.1(1+i)3

Show from the power series (8.1) that ez1·ez2=ez1+z2.

Express the following complex numbers in the x+iyform. 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.

13.(i−3)31−i

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find x,y,r,θ in your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

-1.

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find x,y,r,θ in your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

2-2i.

See all solutions

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