Texte intégral
The Cauchy–Riemann equations come in various equivalent flavours:
1
⎧⎪
⎨⎪
⎩
𝜕𝑢
𝜕𝑥(𝑥0, 𝑦0) = 𝜕𝑣
𝜕𝑦(𝑥0, 𝑦0)
𝜕𝑢
𝜕𝑦(𝑥0, 𝑦0) = −𝜕𝑣
𝜕𝑥(𝑥0, 𝑦0)
And in this case,𝑓′(𝑧0) = 𝜕𝑢
𝜕𝑥(𝑥0, 𝑦0) + 𝑖𝜕𝑣
𝜕𝑥(𝑥0, 𝑦0).
2 Jac𝑓̃(𝑥0, 𝑦0) = (
𝑎 −𝑏
𝑏 𝑎 )for some𝑎, 𝑏 ∈ ℝ. Then𝑓′(𝑧0) = 𝑎 + 𝑖𝑏.
3 𝜕𝑓
𝜕𝑦(𝑧0) = 𝑖𝜕𝑓
𝜕𝑥(𝑧0).
4 𝜕𝑓
𝜕𝑧(𝑥0, 𝑦0) = 0.
Then𝑓′(𝑧0) = 𝜕𝑓
𝜕𝑥(𝑧0) = −𝑖𝜕𝑓
𝜕𝑦(𝑧0) = 𝜕𝑓
𝜕𝑧(𝑧0).
Remember that 𝜕𝑓
𝜕𝑧 ≔12(𝜕𝑓𝜕𝑥+ 𝑖𝜕𝑓𝜕𝑦)and 𝜕𝑓
𝜕𝑧 ≔12(𝜕𝑓𝜕𝑥− 𝑖𝜕𝑓𝜕𝑦).
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