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Mathematics · lesson 1 of 5

Holomorphic functions and Cauchy-Riemann equations

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

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Learn the ideas

Prerequisites: Complex arithmetic and partial derivatives.

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.

Worked examples

Example 1

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

Credit both equations and smoothness of the partials.

u=x²−y² and v=2xy. Then u_x=2x=v_y and u_y=−2y=−v_x.

Example 2

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

f′(z)=3z², so f′(2)=12.

12

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

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

Sample answer and marking guidance

u=x²−y² and v=2xy. Then u_x=2x=v_y and u_y=−2y=−v_x.

Credit both equations and smoothness of the partials.

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

Show answer and explanation

12

f′(z)=3z², so f′(2)=12.

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

Show answer and explanation

No

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 and marking guidance

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

Credit two directions with distinct limiting quotients.

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

Show answer and explanation

A holomorphic function

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.

Show answer and explanation

2

f′(z)=2z, hence 2i.

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

Sample answer and 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.

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 and 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.

Credit the neighborhood and regularity conditions.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.