Question

Evaluate. Write your answer(s) in the form a + bi.

(a) e^(5−9i) / (4π)
(b) log(-3) and Log(-3)

Answer 1

To evaluate e^(5−9i) / (4π), we can use Euler's formula and simplify the expression. To evaluate log(-3) and Log(-3), we use the properties of complex logarithms and find their principal values.

Step-by-step explanation:

To evaluate the expression e^(5−9i) / (4π), we can use Euler's formula, which states that e^(ix) = cos(x) + i sin(x). So, we can rewrite e^(5−9i) as cos(5) + i sin(5) - 9(cos(5) + i sin(5)). To simplify the expression further, we can divide each term by 4π. This gives us the final answer:

(cos(5) - 9cos(5))/(4π) + i(sin(5) - 9sin(5))/(4π)

To evaluate the expressions log(-3) and Log(-3), we need to use the properties of complex logarithms. Since -3 is negative, we can rewrite it in polar form as -3 = 3e^(iπ). The principal value of log(-3) can be found by taking the natural logarithm of the modulus of -3 and adding i times the argument of -3:

log(-3) = ln|3| + iπ

The function Log(-3) represents the multi-valued complex logarithm, so it can take on multiple values. One of the values is the principal value we found earlier (ln|3| + iπ), and the other values can be obtained by adding or subtracting multiples of 2πi.