Just Solving Questions

 Find all the roots of (1+z)^5=(1-z)^5 | Complex Analysis 1. Simplify the equation: Divide both sides by (1 - z)^5: (1 + z)^5 / (1 - z)^5 = 1 Recognize this as a special case: This equation is of the form (a/b)^5 = 1, where a = 1 + z and b = 1 - z. 2. Find the fifth roots of unity: The equation (a/b)^5 = 1 implies that a/b is a fifth root of unity. The fifth roots of unity are given by: e^(2kπi/5), where k = 0, 1, 2, 3, 4 3. Express a/b in terms of fifth roots of unity: a/b = e^(2kπi/5) 4. Solve for z: Substitute a = 1 + z and b = 1 - z: (1 + z) / (1 - z) = e^(2kπi/5) Cross-multiply: 1 + z = e^(2kπi/5) * (1 - z) Expand and rearrange: z + z*e^(2kπi/5) = e^(2kπi/5) - 1 Factor out z: z * (1 + e^(2kπi/5)) = e^(2kπi/5) - 1 Solve for z: z = (e^(2kπi/5) - 1) / (1 + e^(2kπi/5)) 5. Simplify the expression for z: Multiply the numerator and denominator by e^(-kπi/5): z = (e^(kπi/5) - e^(-kπi/5)) / (e^(kπi/5) + e^(-kπi/5)) Use Euler's formula: z = (2i sin(kπ/...