Cauchy riemann equations

1. CAUCHY-RIEMANNEQUATION
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2. Cauchy-RiemannEquation
푓푧is holomorphic ⇒ 휕푓 휕푥 = 휕푓 푖휕y

3. Cauchy-RiemannEquation
푓푧is holomorphic ⟺ 휕푓 휕푥 = 휕푓 푖휕y

4. Cauchy-RiemannEquation
푓푧is holomorphic ⟺ 휕푓 휕푥 = 휕푓 푖휕y

5. Cauchy-RiemannEquation 휕푓 휕 푧 = 휕푓 휕푥 휕푥 휕 푧 + 휕푓 휕푦 휕푦 휕 푧

6. Cauchy-RiemannEquation 휕푓 휕 푧 = 휕푓 휕푥 휕푥 휕 푧 + 휕푓 휕푦 휕푦 휕 푧

7. Cauchy-RiemannEquation 휕푓 휕 푧 = 휕푓 휕푥 휕푥 휕 푧 + 휕푓 휕푦 휕푦 휕 푧

8. Cauchy-RiemannEquation 휕푓 휕 푧 = 휕푓 휕푥 12+ 휕푓 휕푦 푖 2
∴푥= 푧+ 푧 2,푦= 푧− 푧 2푖

9. Cauchy-RiemannEquation 휕푓 휕 푧 = 휕푓 휕푥 12+ 휕푓 휕푦 푖 2
∴푥= 푧+ 푧 2,푦= 푧− 푧 2푖

10. Cauchy-RiemannEquation 휕푓 휕 푧 = 12 휕푓 휕푥 +푖 휕푓 휕푦
∴푥= 푧+ 푧 2,푦= 푧− 푧 2푖

11. Cauchy-RiemannEquation
푓푧is holomorphic⟺ 휕푓 휕푥 = 휕푓 푖휕y
⇕ 휕푓 휕 푧 = 12 휕푓 휕푥 +푖 휕푓 휕푦 = 12 휕푓 푖휕푦 +푖 휕푓 휕푦 =0

12. Cauchy-RiemannEquation
푓푧is holomorphic⟺ 휕푓 휕푥 = 휕푓 푖휕y
⇕ 휕푓 휕 푧 = 12 휕푓 휕푥 +푖 휕푓 휕푦 = 12 휕푓 푖휕푦 +푖 휕푓 휕푦 =0

13. Cauchy-RiemannEquation
푓푧is holomorphic⟺ 휕푓 휕푥 = 휕푓 푖휕y
⇕ 휕푓 휕 푧 = 12 휕푓 휕푥 +푖 휕푓 휕푦 = 12 휕푓 푖휕푦 +푖 휕푓 휕푦 =0

14. Cauchy-RiemannEquation
푓푧is holomorphic⟺ 휕푓 휕푥 = 휕푓 푖휕y
⇕ 휕푓 휕 푧 = 12 휕푓 휕푥 +푖 휕푓 휕푦 = 12 휕푓 푖휕푦 +푖 휕푓 휕푦 =0

15. Cauchy-RiemannEquation
푓푧is holomorphic⟺ 휕푓 휕푥 = 휕푓 푖휕y
⇕ 휕푓 휕 푧 = 12 휕푓 휕푥 +푖 휕푓 휕푦 = 12−푖 휕푓 휕푦 +푖 휕푓 휕푦 =0

16. Cauchy-RiemannEquation
∴푓푧is holomorphic ⇔ 휕푓 휕 푧 =0
푉푒푟푦푠푖푚푝푙푒!

17. Cauchy-RiemannEquation
∴푓푧is holomorphic ⇔ 휕푓 휕 푧 =0
푉푒푟푦푠푖푚푝푙푒!

18. Exercise
Is푧3holomorphic function?
Answer
푓푧=푧3=푧 푧 32⇒ 휕푓 휕 푧 = 32 푧 32 푧≠0
∴푧3is not holomorphic function.

19. Exercise
Is푧3holomorphic function?
Answer
푓푧=푧3=푧 푧 32⇒ 휕푓 휕 푧 = 32 푧 32 푧≠0
∴푧3is not holomorphic function.

20. Exercise
Is푧3holomorphic function?
Answer
푓푧=푧3=푧 푧 32⇒ 휕푓 휕 푧 = 32 푧 32 푧≠0
∴푧3is not holomorphic function.

21. Exercise
Is푧3holomorphic function?
Answer
푓푧=푧3=푧 푧 32⇒ 휕푓 휕 푧 = 32 푧 32 푧≠0
∴푧3is not holomorphic function.

22. END