CAUCHY-RIEMANNEQUATION

Hanpen Robot

Cauchy-Riemann Equation

𝑓 𝑧 is holomorphic ⇒𝜕𝑓

𝜕𝑥=

𝜕𝑓

𝑖𝜕y

Cauchy-Riemann Equation

𝑓 𝑧 is holomorphic ⟺𝜕𝑓

𝜕𝑥=

𝜕𝑓

𝑖𝜕y

Cauchy-Riemann Equation

𝑓 𝑧 is holomorphic ⟺𝜕𝑓

𝜕𝑥=

𝜕𝑓

𝑖𝜕y

Cauchy-Riemann Equation

𝜕𝑓

𝜕 𝑧=𝜕𝑓

𝜕𝑥

𝜕𝑥

𝜕 𝑧+𝜕𝑓

𝜕𝑦

𝜕𝑦

𝜕 𝑧

Cauchy-Riemann Equation

𝜕𝑓

𝜕 𝑧=𝜕𝑓

𝜕𝑥

𝜕𝑥

𝜕 𝑧+𝜕𝑓

𝜕𝑦

𝜕𝑦

𝜕 𝑧

Cauchy-Riemann Equation

𝜕𝑓

𝜕 𝑧=𝜕𝑓

𝜕𝑥

𝜕𝑥

𝜕 𝑧+𝜕𝑓

𝜕𝑦

𝜕𝑦

𝜕 𝑧

Cauchy-Riemann Equation

𝜕𝑓

𝜕 𝑧=𝜕𝑓

𝜕𝑥

1

2+𝜕𝑓

𝜕𝑦

𝑖

2

∴ 𝑥 =𝑧 + 𝑧

2, 𝑦 =

𝑧 − 𝑧

2𝑖

Cauchy-Riemann Equation

𝜕𝑓

𝜕 𝑧=𝜕𝑓

𝜕𝑥

1

2+𝜕𝑓

𝜕𝑦

𝑖

2

∴ 𝑥 =𝑧 + 𝑧

2, 𝑦 =

𝑧 − 𝑧

2𝑖

Cauchy-Riemann Equation

𝜕𝑓

𝜕 𝑧=1

2

𝜕𝑓

𝜕𝑥+ 𝑖

𝜕𝑓

𝜕𝑦

∴ 𝑥 =𝑧 + 𝑧

2, 𝑦 =

𝑧 − 𝑧

2𝑖

Cauchy-Riemann Equation

𝑓 𝑧 is holomorphic ⟺𝜕𝑓

𝜕𝑥=

𝜕𝑓

𝑖𝜕y

⇕

𝜕𝑓

𝜕 𝑧=1

2

𝜕𝑓

𝜕𝑥+ 𝑖

𝜕𝑓

𝜕𝑦=1

2

𝜕𝑓

𝑖𝜕𝑦+ 𝑖

𝜕𝑓

𝜕𝑦= 0

Cauchy-Riemann Equation

𝑓 𝑧 is holomorphic ⟺𝜕𝑓

𝜕𝑥=

𝜕𝑓

𝑖𝜕y

⇕

𝜕𝑓

𝜕 𝑧=1

2

𝜕𝑓

𝜕𝑥+ 𝑖

𝜕𝑓

𝜕𝑦=1

2

𝜕𝑓

𝑖𝜕𝑦+ 𝑖

𝜕𝑓

𝜕𝑦= 0

Cauchy-Riemann Equation

𝑓 𝑧 is holomorphic ⟺𝜕𝑓

𝜕𝑥=

𝜕𝑓

𝑖𝜕y

⇕

𝜕𝑓

𝜕 𝑧=1

2

𝜕𝑓

𝜕𝑥+ 𝑖

𝜕𝑓

𝜕𝑦=1

2

𝜕𝑓

𝑖𝜕𝑦+ 𝑖

𝜕𝑓

𝜕𝑦= 0

Cauchy-Riemann Equation

𝑓 𝑧 is holomorphic ⟺𝜕𝑓

𝜕𝑥=

𝜕𝑓

𝑖𝜕y

⇕

𝜕𝑓

𝜕 𝑧=1

2

𝜕𝑓

𝜕𝑥+ 𝑖

𝜕𝑓

𝜕𝑦=1

2

𝜕𝑓

𝑖𝜕𝑦+ 𝑖

𝜕𝑓

𝜕𝑦= 0

Cauchy-Riemann Equation

𝑓 𝑧 is holomorphic ⟺𝜕𝑓

𝜕𝑥=

𝜕𝑓

𝑖𝜕y

⇕

𝜕𝑓

𝜕 𝑧=1

2

𝜕𝑓

𝜕𝑥+ 𝑖

𝜕𝑓

𝜕𝑦=1

2−𝑖

𝜕𝑓

𝜕𝑦+ 𝑖

𝜕𝑓

𝜕𝑦= 0

Cauchy-Riemann Equation

∴ 𝑓 𝑧 is holomorphic ⇔𝜕𝑓

𝜕 𝑧= 0

𝑉𝑒𝑟𝑦 𝑠𝑖𝑚𝑝𝑙𝑒!

Cauchy-Riemann Equation

∴ 𝑓 𝑧 is holomorphic ⇔𝜕𝑓

𝜕 𝑧= 0

𝑉𝑒𝑟𝑦 𝑠𝑖𝑚𝑝𝑙𝑒!

Exercise

Is 𝑧 3 holomorphic function?

Answer

𝑓 𝑧 = 𝑧 3 = 𝑧 𝑧32 ⇒

𝜕𝑓

𝜕 𝑧=3

2𝑧32 𝑧 ≠ 0

∴ 𝑧 3 is not holomorphic function.

Exercise

Is 𝑧 3 holomorphic function?

Answer

𝑓 𝑧 = 𝑧 3 = 𝑧 𝑧32 ⇒

𝜕𝑓

𝜕 𝑧=3

2𝑧32 𝑧 ≠ 0

∴ 𝑧 3 is not holomorphic function.

Exercise

Is 𝑧 3 holomorphic function?

Answer

𝑓 𝑧 = 𝑧 3 = 𝑧 𝑧32 ⇒

𝜕𝑓

𝜕 𝑧=3

2𝑧32 𝑧 ≠ 0

∴ 𝑧 3 is not holomorphic function.

Exercise

Is 𝑧 3 holomorphic function?

Answer

𝑓 𝑧 = 𝑧 3 = 𝑧 𝑧32 ⇒

𝜕𝑓

𝜕 𝑧=3

2𝑧32 𝑧 ≠ 0

∴ 𝑧 3 is not holomorphic function.

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