# Holomorphic functions and Cauchy-Riemann equations

> A derivative must have the same limit along every complex direction.

Course: [Advanced mathematical methods](https://studybrix.com/en/courses/advanced-mathematical-methods/index.md)
Level: advanced
Prerequisites: Complex arithmetic and partial derivatives.

## Notes

### Complex differentiability

A derivative must have the same limit along every complex direction.

### Cauchy-Riemann

For f=u+iv, differentiability implies u_x=v_y and u_y=−v_x when the partials exist.

### Sufficient hypotheses

Continuous partial derivatives satisfying the equations on a neighborhood give holomorphicity there.

### Not a point test

Equations at one isolated point alone do not prove holomorphicity throughout a neighborhood.

## Practice questions

### 1. Verify the Cauchy-Riemann equations for f(z)=z².

Sample answer / marking guidance: u=x²−y² and v=2xy. Then u_x=2x=v_y and u_y=−2y=−v_x.

Explanation: Credit both equations and smoothness of the partials.

### 2. For f(z)=z³, find the real derivative value f′(2).

Answer: 12

Explanation: f′(z)=3z², so f′(2)=12.

### 3. Is complex conjugation f(z)=conjugate(z) holomorphic on an open subset of C?

A. It is a polynomial in z
B. No
C. Only because it is continuous
D. Yes everywhere

Answer: No

Explanation: For u=x,v=−y, u_x=1 but v_y=−1, so the Cauchy-Riemann equations fail.

### 4. Use directional difference quotients to show conjugation is not complex differentiable at zero.

Sample answer / marking guidance: Along real h, conjugate(h)/h=1; along imaginary h, the quotient is −1. The limits disagree.

Explanation: Credit two directions with distinct limiting quotients.

### 5. What does a power series define inside its radius of convergence?

A. A function with no derivative
B. A necessarily constant function
C. Only a function on the real axis
D. A holomorphic function

Answer: A holomorphic function

Explanation: Complex power series can be differentiated termwise inside their convergence disk.

### 6. For f(z)=z² at z=i, write f′(i)=a i and find a.

Answer: 2

Explanation: f′(z)=2z, hence 2i.

### 7. Why is real differentiability of a map R²→R² weaker than complex differentiability?

Sample answer / marking guidance: A real derivative may be any real linear map, whereas a complex derivative must act like multiplication by a single complex number, imposing the Cauchy-Riemann structure.

Explanation: Credit the restriction on the derivative map.

### 8. State a sufficient local condition using Cauchy-Riemann equations that avoids the isolated-point trap.

Sample answer / marking guidance: If u and v have continuous first partials on an open neighborhood and satisfy the equations throughout it, then f=u+iv is holomorphic there.

Explanation: Credit the neighborhood and regularity conditions.

## Further reading

- https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
- https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.