Exponential function of a complex number question?

In summary, the given values for a) and b) are 5 and 5PI/6 respectively, and using the property of complex numbers, e^(a+bi) can be simplified to e^5 (-sqrt(3)/2 + i/2). Further steps are not necessary.
  • #1
Mandynash
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Homework Statement



The question is located here http://i51.tinypic.com/nex2q1.jpg

Homework Equations



The value I have been given for a) is 5
The value I have been given for b) is 5PI/6

The Attempt at a Solution



Note that e^(a + bi) = e^a e^(bi) = e^a (cos b + i sin b).
(e^a is the modulus, and b is the argument, because |e^(bi)| = sqrt(cos^2(b) + sin^2(b)) = 1.)

Letting a = 5 and b = 5π/6 yields
e^(a+bi) = e^5 (cos 5π/6 + i sin 5π/6) = e^5 (-sqrt(3)/2 + i/2).

No idea what else I need to do! thanks in advance for helping
 
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  • #2
Looks good!
 

FAQ: Exponential function of a complex number question?

What is the definition of an exponential function of a complex number?

An exponential function of a complex number is a function of the form f(z) = e^z, where z is a complex number and e is the base of the natural logarithm. This function is used to describe the growth or decay of a quantity that involves complex numbers.

How do you calculate the value of an exponential function of a complex number?

To calculate the value of an exponential function of a complex number, you can use the formula f(z) = e^z = e^(x+iy) = e^x * e^(iy) = e^x * (cos(y) + i*sin(y)), where x is the real part of z and y is the imaginary part of z.

What are the properties of an exponential function of a complex number?

Some properties of an exponential function of a complex number include:

  • The value of the function is always a complex number.
  • The function is differentiable and its derivative is equal to itself.
  • The exponential function of a sum of two complex numbers is equal to the product of the exponential functions of each individual number.
  • The exponential function of a complex number raised to a power is equal to the exponential function of the number multiplied by that power.

How is the exponential function of a complex number related to the trigonometric functions?

The exponential function of a complex number is related to the trigonometric functions through Euler's formula, e^(iy) = cos(y) + i*sin(y). This relationship allows for the use of complex numbers in trigonometric calculations and vice versa.

What are some real-life applications of exponential functions of complex numbers?

Exponential functions of complex numbers are used in various fields of science and engineering, such as physics, electrical engineering, and signal processing. They are also used in finance and economics to model growth and decay of investments and populations. Additionally, they have applications in fluid dynamics, quantum mechanics, and image processing.

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