Complex Logarithm Homework: Find -Ln(1-e(iθ))

In summary, the problem is to find -Ln(1-e(i\theta) in terms of theta. The equation ln(z) = ln(rei\theta) = Ln(r) + ln(ei\theta) = Ln(r) + i\theta is used to break the logarithm into real and complex parts. However, two possible solutions result in either an infinite value or only an imaginary part, which does not match the given problem. To solve this, the expression 1 - eiθ must be expressed in Cartesian form.
  • #1
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Homework Statement


find -Ln(1-e(i[tex]\theta[/tex]) (in terms of theta)

(this is me just skipping the part of the problem I know and going straight to what I can't figure out)

Homework Equations


ln(z) = ln(rei[tex]\theta[/tex])=Ln(r) + ln(ei[tex]\theta[/tex]) = Ln(r) + i[tex]\theta[/tex]

The Attempt at a Solution


I don't really know how to break this logarithm up into real and complex parts, the two ways I considered were

= -(Ln(1-1) + i[tex]\theta[/tex])
but that ends up with Ln(0) which blows up to infinity and doesn't make sense in this problem.

= -(Ln(1) + i[tex]\theta[/tex])
but this is just -i[tex]\theta[/tex] which leaves only an imaginary part, and the problem implies there is a real logarithmic solution as well.
 
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  • #2
hi soothsayer! :smile:

(have a theta: θ :wink:)

you need to express 1 - e in Cartesian form, a + ib :wink:
 

1. What is a complex logarithm?

A complex logarithm is a mathematical function that maps a complex number to another complex number. It is defined as the inverse function of the exponential function and is denoted as log z or ln z.

2. How do you find the complex logarithm of a number?

To find the complex logarithm of a number, you can use the formula log z = ln|z| + iθ, where |z| is the absolute value of the number and θ is the argument (or angle) of the number in polar form. Alternatively, you can use a calculator or computer program to find the complex logarithm.

3. What is the complex logarithm of a negative number?

The complex logarithm of a negative number is not a real number. Instead, it is a complex number with a real part and an imaginary part. In other words, it is a number in the form a + bi, where a and b are real numbers and i is the imaginary unit. It is important to note that the complex logarithm of a negative number is not a unique value, as there are infinitely many values that satisfy the equation e^z = -x.

4. How do you find the complex logarithm of a complex number?

To find the complex logarithm of a complex number, you can use the same formula as for finding the logarithm of a real number (log z = ln|z| + iθ). However, you must first convert the complex number to polar form (r(cosθ + isinθ)) and then plug in the values for r and θ into the formula. Alternatively, you can use a calculator or computer program to find the complex logarithm.

5. What is the complex logarithm of 0?

The complex logarithm of 0 is undefined. This is because there is no complex number z that satisfies the equation e^z = 0. In other words, there is no complex number that can be raised to a power to give 0 as a result.

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