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Chapter 2: Q15P (page 71)

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

siniz = i sin z

Short Answer

Expert verified

The equation sinh iz = i sin zis verified using the equations (11.4), (12.2) and (12.3).

Step by step solution

01

Given Information

Givenequationis sinh iz= i sin z

02

Definition of Hyperbolic Function.

The term "Hyperbolic Function" refers to the relationship between a point on a hyperbola's distance from its origin and its coordinate axes, expressed as a function of an angle.

03

 Use exponential form to expand the equation

Given the equation is sinh iz = isin z.

The exponential form of the given equation is,

sin(z)=ez-e-z2i.

鈥�.(1)

Let z鈫�zithan sin (iz)=ezi.i-e-zi.i2i.

Solve sin (iz) to prove the given equation.

siniz=ezi-e-zi2i

Multiply numerator and denominator by i.

sinhiz=ii脳expzi-e-zi2sinhiz=iexpzi-exp-zi2isinhiz=ii脳expzi-e-zi2sinhiz=isin(z)

Hence, the equation is verified.

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Most popular questions from this chapter

Test each of the following series for convergence.

鈭�(1+i)n

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鈥攖ry 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.

8.2e5蟺颈/6

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鈥攖ry 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.

10.role="math" localid="1653071800850" ei蟺+e-i蟺

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,胃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.

4(cos2蟺3+isin2蟺3).

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.

253+4i.

See all solutions

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