Question

Consider the equation e²⁻ᶻ=-1+i. - Find all the solutions to the equation based on the definition of complex exponential function.

Answer 1

To find the solutions to the equation e²⁻ᶻ=-1+i, we can rewrite it in the form e²⁻ᶻ+1-i=0 and solve it using the quadratic formula.

Step-by-step explanation:

To find the solutions to the equation e²⁻ᶻ=-1+i, we can rewrite it in the form e²⁻ᶻ+1-i=0. This is a quadratic equation in terms of eᶻ, so we can solve it using the quadratic formula. Letting a = 1, b = -1, and c = -1, we have:

eᶻ = (-b ± √(b² - 4ac)) / (2a)

Substituting the values, we get:

eᶻ = (1 ± √(1 - 4(-1)(-1))) / 2 = (1 ± √(-3)) / 2

Since eᶻ represents the exponential function, we can express it in polar form as eᶻ = r * (cosθ + isinθ), where r is the magnitude and θ is the argument. Therefore, the solutions to the equation are:

eᶻ = (1 ± √(3)i) / 2