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 = (2isin(kπ/5)) / (2cos(kπ/5))

• Simplify: z = i*tan(kπ/5)

Therefore, the roots of the equation (1 + z)^5 = (1 - z)^5 are:

• z = i*tan(kπ/5), where k = 0, 1, 2, 3, 4

Note:

• When k = 0, z = 0.
• The other four roots are non-zero complex numbers.