the cauchy–riemann equations · 2020. 10. 5. · the cauchy–riemann equations the...

8

Upload: others

Post on 08-Mar-2021

11 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: The Cauchy–Riemann equations · 2020. 10. 5. · The Cauchy–Riemann equations The Cauchy–Riemann equations come in various equivalent flavours: 1 ⎧ ⎪ ⎨ ⎪ ⎩ 𝜕𝑢
Page 2: The Cauchy–Riemann equations · 2020. 10. 5. · The Cauchy–Riemann equations The Cauchy–Riemann equations come in various equivalent flavours: 1 ⎧ ⎪ ⎨ ⎪ ⎩ 𝜕𝑢
Page 3: The Cauchy–Riemann equations · 2020. 10. 5. · The Cauchy–Riemann equations The Cauchy–Riemann equations come in various equivalent flavours: 1 ⎧ ⎪ ⎨ ⎪ ⎩ 𝜕𝑢

The Cauchy–Riemann equationsThe 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 Jac ̃𝑓 (𝑥0, 𝑦0) = (𝑎 −𝑏𝑏 𝑎 ) for some 𝑎, 𝑏 ∈ ℝ. Then 𝑓 ′(𝑧0) = 𝑎 + 𝑖𝑏.

3𝜕𝑓𝜕𝑦 (𝑧0) = 𝑖𝜕𝑓

𝜕𝑥 (𝑧0).

4𝜕𝑓𝜕𝑧

(𝑥0, 𝑦0) = 0.

Then 𝑓 ′(𝑧0) = 𝜕𝑓𝜕𝑥 (𝑧0) = −𝑖𝜕𝑓

𝜕𝑦 (𝑧0) = 𝜕𝑓𝜕𝑧 (𝑧0).

Remember that 𝜕𝑓𝜕𝑧 ≔ 1

2 (𝜕𝑓𝜕𝑥 + 𝑖 𝜕𝑓

𝜕𝑦 ) and 𝜕𝑓𝜕𝑧 ≔ 1

2 (𝜕𝑓𝜕𝑥 − 𝑖 𝜕𝑓

𝜕𝑦 ).

Page 4: The Cauchy–Riemann equations · 2020. 10. 5. · The Cauchy–Riemann equations The Cauchy–Riemann equations come in various equivalent flavours: 1 ⎧ ⎪ ⎨ ⎪ ⎩ 𝜕𝑢
Page 5: The Cauchy–Riemann equations · 2020. 10. 5. · The Cauchy–Riemann equations The Cauchy–Riemann equations come in various equivalent flavours: 1 ⎧ ⎪ ⎨ ⎪ ⎩ 𝜕𝑢
Page 6: The Cauchy–Riemann equations · 2020. 10. 5. · The Cauchy–Riemann equations The Cauchy–Riemann equations come in various equivalent flavours: 1 ⎧ ⎪ ⎨ ⎪ ⎩ 𝜕𝑢
Page 7: The Cauchy–Riemann equations · 2020. 10. 5. · The Cauchy–Riemann equations The Cauchy–Riemann equations come in various equivalent flavours: 1 ⎧ ⎪ ⎨ ⎪ ⎩ 𝜕𝑢
Page 8: The Cauchy–Riemann equations · 2020. 10. 5. · The Cauchy–Riemann equations The Cauchy–Riemann equations come in various equivalent flavours: 1 ⎧ ⎪ ⎨ ⎪ ⎩ 𝜕𝑢