The document summarizes the solution steps to solving the complex integral C dz / (1 + z^2) = 0 from z = 1 to z = e^iθ (0 ≤ θ < 2π). Making a substitution of z = e^iθ and evaluating the integral from 0 to 2π using trigonometric identities results in the solution of 0.

1. 𝐶
𝑑𝑧
1 + 𝑧2
=
0
2𝜋
𝑖𝑒 𝑖𝜃
1 + 𝑒2𝑖𝜃
𝑑𝜃 =
0
2𝜋
𝑖𝑑𝜃
𝑒−𝑖𝜃 + 𝑒 𝑖𝜃
𝐶: 𝑧 = 1 ⇔ 𝑧 = 𝑒 𝑖𝜃 (0 ≤ 𝜃 < 2𝜋)
𝑑𝑧
𝑑𝜃
= 𝑖𝑒 𝑖𝜃 ∴ 𝑑𝑧 = 𝑖𝑒 𝑖𝜃 𝑑𝜃
2015/04/29 Hanpen Robot
=
𝑖
2 0
2𝜋
𝑑𝜃
𝑐𝑜𝑠𝜃
=
𝑖
2
log 𝑒 𝑡𝑎𝑛
𝜃
2
+
𝜋
4 0
2𝜋
∵ 𝑐𝑜𝑠𝜃 =
𝑒−𝑖𝜃+𝑒 𝑖𝜃
2

2. =
𝑖
2
log 𝑒 𝑡𝑎𝑛
𝜃
2
+
𝜋
4 0
2𝜋
=
𝑖
2
log 𝑒 𝑡𝑎𝑛
5
4
𝜋 − log 𝑒 𝑡𝑎𝑛
1
4
𝜋
=
𝑖
2
log 𝑒 1 − log 𝑒 1 = 0
2015/04/29 Hanpen Robot

3. 𝑑𝜃
𝑐𝑜𝑠𝜃
の計算方法は以下を見てください
https://www.youtube.com/watch?v=ZcX8XQWxWVA
2015/04/29 Hanpen Robot