Mathematics.

complex differentiation

Complex Differentiation

Complex Analysis25 minDifficulty5 out of 10

You should know: cauchy riemann

Overview

The complex derivative of f at z0 is defined by the same difference-quotient limit as in single-variable real calculus, f'(z0) = lim_{h→0} [f(z0+h) − f(z0)]/h, except now h ranges over complex numbers approaching 0 from every direction in the plane. Because the limit must agree regardless of the direction of approach, complex differentiability at a point is a far more demanding condition than real differentiability, and it is captured algebraically by the Cauchy–Riemann equations. All the familiar rules — sum, product, quotient, and chain rules — carry over unchanged from real calculus, so polynomials, exponentials, and compositions of differentiable functions are differentiated exactly as expected. The theory built on top of this single definition (analyticity, Cauchy's theorems, residues) is what gives complex analysis its remarkable power and rigidity compared to real analysis.

Intuition

The key subtlety of complex differentiation is that h → 0 in the complex plane means h can approach 0 along infinitely many directions — along the real axis, the imaginary axis, or any spiral — and the definition demands the same limiting ratio no matter which path is taken. In real calculus h can only approach from the left or the right, so agreement between two directions is already a meaningful constraint; in the complex case, agreement across an entire continuum of directions is what makes the condition so restrictive, and it is exactly this restriction that the Cauchy-Riemann equations encode algebraically.

Formal Definition

Definition

For f defined on an open set containing z0, the complex derivative is:

f′(z0)=lim⁡h→0f(z0+h)−f(z0)h,h∈Cf'(z_0) = \lim_{h \to 0} \frac{f(z_0+h) - f(z_0)}{h}, \qquad h \in \mathbb{C}
Definition of the complex derivative
(f+g)′=f′+g′,(fg)′=f′g+fg′,(fg)′=f′g−fg′g2(f+g)' = f'+g', \qquad (fg)' = f'g + fg', \qquad \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}
Sum, product, and quotient rules
(f∘g)′(z)=f′(g(z)) g′(z)(f\circ g)'(z) = f'(g(z))\,g'(z)
Chain rule

Worked Examples

  1. Apply the power rule term by term, exactly as in real calculus: d/dz(z^n) = n z^{n-1}.

    f′(z)=3z2+2f'(z) = 3z^2 + 2

  2. Evaluate at z = i, using i² = -1.

    f′(i)=3(i)2+2=3(−1)+2=−3+2=−1f'(i) = 3(i)^2 + 2 = 3(-1)+2 = -3+2 = -1

Answer: f'(z) = 3z² + 2, and f'(i) = -1.

Practice Problems

Difficulty 4/10

Find f'(z) for f(z) = (z² + 1)/(z - 2) using the quotient rule, for z ≠ 2.

Difficulty 5/10

Differentiate f(z) = sin(z²) with respect to z.

Quiz

The complex derivative f'(z0) is defined via a limit where h approaches 0:
Which of the following rules does NOT carry over unchanged from real to complex differentiation?
For f(z) = z^4, f'(z) equals:

Summary

  • The complex derivative is defined by the same difference-quotient limit as in real calculus, but h ranges over all complex directions, making the condition much stronger.
  • Sum, product, quotient, and chain rules for differentiation carry over unchanged from real calculus.
  • Direction-independence of the limit is precisely what the Cauchy-Riemann equations enforce algebraically.

References

  1. WebsiteWikipedia — Holomorphic function