Contour Integration

Draw paths and integrate along them - Cauchy's theorems come alive

Contour Integration

In the complex plane there are endless paths between two points, and an integral can depend on which one you take. For analytic functions, it doesn't.

1. Contour Integrator

Click on the canvas to draw a custom contour, or select a preset shape. Then choose a function to integrate along your path. Try 1/z around a circle that contains the origin.

💡 Observations:

  • For closed contours (start = end), try integrating f(z) = z - notice it gives 0!
  • Now try f(z) = 1/z around a circle containing the origin - you get 2πi (non-zero!)
  • The rectangle and circle give the same result whenever they enclose the same singularities - closed-contour integrals depend only on the enclosed poles, not on the shape of the path
  • Custom paths let you explore more complex contours - try drawing a figure-eight or spiral!
Go deeper:Contour Integration

One of the most powerful tools in complex analysis is the ability to integrate functions along contours (paths) in the complex plane. Unlike real integration, complex integrals depend on the path taken - unless the function is analytic!

The fundamental results - Cauchy's Integral Theorem and Cauchy's Integral Formula - connect analyticity to path independence and allow us to evaluate integrals that would be impossible in real analysis.

The contour integral of f(z) along a path γ is defined as:

∫γ f(z) dz = ∫ab f(γ(t)) · γ'(t) dt

2. Cauchy's Integral Theorem

Choose a function and a closed contour. Analytic functions give 0 every time; watch what changes when a pole sits inside the contour.

Square: f(z) = z²
Analyticity: Everywhere (entire function)

Domain Coloring Visualization:

The selected contour encloses the plotted region. Zeros appear black, poles appear white.

Contour Integral Result:

∮γ z² dz =
-0.0003 - 0.0106i
Magnitude: 0.0106
Expected Result:
0 (Cauchy's Theorem applies)

🔍 Cauchy's Integral Theorem Explained:

  • If f is analytic everywhere inside and on a closed contour γ, then ∮γ f(z) dz = 0
  • This is why z, z², e^z, and sin(z) all give 0 - they're analytic everywhere (entire functions)
  • f(z) = 1/z is NOT analytic at z = 0. If the contour encloses the origin, the integral is 2πi (non-zero!)
  • If the pole is outside the contour, the function is analytic inside, so the integral is 0
  • Non-analytic functions like z̄ violate Cauchy's theorem - integrals are generally non-zero
Go deeper:Cauchy's Integral Theorem

One of the most important theorems in complex analysis:

Cauchy's Integral Theorem
If f is analytic in a simply connected domain D, and γ is any closed contour in D, then:
∮γ f(z) dz = 0

This means that for analytic functions, closed path integrals always equal zero! This is not true for non-analytic functions.

3. Cauchy's Integral Formula

Move the test point a and compare f(a) computed directly with the value recovered from the contour integral. They match while a stays inside the contour.

Test Point: a = 0.50 + 0.30i

✓ Point is inside the contour (formula applies)

Circular contour centered at origin with radius 1.5

Verification of Cauchy's Integral Formula:

Direct Evaluation
f(a) = f(0.50+0.30i) = z²
0.1600 + 0.3000i
Cauchy's Formula
(1/2πi) ∮ f(z)/(z-a) dz
0.1584 + 0.2996i
✓ Formula Verified!
Absolute Error: -0.001563 - 0.000417i
Error Magnitude: 0.001618
Relative Error: 0.48%
The formula accurately recovers f(a) from the contour integral!

🎯 Cauchy's Integral Formula Explained:

  • This formula is remarkable: it says the value of f at any interior point is completely determined by its values on the boundary
  • No such result exists in real analysis! This is unique to complex analytic functions.
  • The formula implies that analytic functions are infinitely differentiable (you can differentiate under the integral)
  • Generalized form: f(n)(a) = (n!/2πi) ∮ f(z)/(z-a)n+1 dz gives all derivatives!
  • This leads to power series representations (Taylor series) for all analytic functions
Go deeper:Cauchy's Integral Formula

An even more remarkable result - we can recover the value of an analytic function at any point from its values on a surrounding contour:

Cauchy's Integral Formula
If f is analytic inside and on a simple closed contour γ, and a is inside γ, then:
f(a) = (1/2πi) ∮γ f(z)/(z-a) dz

This formula is extraordinary: knowing f on the boundary completely determines f inside! This has no analog in real analysis.

4. Path Independence

Bend the second path with the curvature slider and compare the two integrals. For analytic functions they agree; try the conjugate or 1/z to see them split.

Adjust how much the curved path (purple dashed) deviates from the straight path (blue solid)

Path 1 (Straight Line)
Direct path from start to end
∫Path 1 z² dz =
-0.2084 + 3.5348i
Path 2 (Curved Path)
Curved path with curvature 1.0
∫Path 2 z² dz =
-0.2210 + 3.5329i
✗ PATH DEPENDENT
Difference between paths:
0.0126 + 0.0019i
Magnitude: 0.012744
⚠ Small numerical error - analytically these should be equal

🔄 Path Independence Explained:

  • For analytic functions (z, z², e^z, sin(z)), the integral between two points is the same regardless of the path taken
  • This is similar to conservative vector fields in real calculus (∇f has path-independent line integrals)
  • For non-analytic functions (z̄) or functions with singularities between paths (1/z), different paths give different integrals
  • Path independence is equivalent to saying ∮ f(z) dz = 0 for all closed curves (Cauchy's theorem)
  • Try increasing curvature to make paths more different - analytic functions still give same result!
Go deeper:Path Independence

For analytic functions, the integral between two points is path independent - it doesn't matter which path you take! This is similar to conservative vector fields in real analysis.

If f is analytic in a simply connected domain D, and γ₁, γ₂ are any two paths from a to b in D, then:
∫γ₁ f(z) dz = ∫γ₂ f(z) dz

For non-analytic functions, this is false - different paths give different integrals.

Key Takeaways

  • A contour integral adds up f(z) dz along a path in the plane
  • Cauchy's Theorem: an analytic function integrates to 0 around a closed contour
  • Cauchy's Formula: the values on a contour determine f everywhere inside
  • For analytic functions, the integral between two points ignores the path

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