Complex Analysis: Part 26 - Keyhole Contour and Contour Integrals

Introduction

• Continuation of complex analysis, focusing on contour integrals.
• Aim to understand a special contour integral.
• Precursor to proving Cauchy's Integral Formula in the next video.
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Recap of Previous Video

• Generalization of Cauchy's theorem for specific domains.

Keyhole Contour

• Discussion focuses on curves with a specific shape, often called a "keyhole contour."
• Such contours are used to explore functions defined holomorphically with exception points.
• Example function: ( \frac{1}{z} ).

Contour Integral in Keyhole Domain

• Consider a function ( G ) holomorphic on a disk with an exception at zero.
• Study the contour integral along a keyhole-shaped curve.
• Cauchy's Theorem: Integral along the curve equals zero regardless of curve orientation.

Parameters

• Introduce radius of inner circle (( \epsilon )) and width of keyhole (( \Delta )).
• Outer radius is fixed, contour named ( \gamma_{\epsilon, \Delta} ).

Analysis of Contour Integrals

• Split integral into four parts: outer circle, two lines, inner circle.
• Parts named ( \gamma_1, \gamma_2, \gamma_3, \gamma_4 ).

Process of Analysis

1. Outer Circle (( \gamma_1 ))

   • When ( \Delta \to 0 ), ( \gamma_1 ) becomes full circle ( \Gamma_1 ).
   • Absolute value estimate reveals that integral approaches zero as curve length goes to zero.

2. Inner Circle (( \gamma_3 ))

   • Similar argument as outer circle.
   • Orientation is opposite; index ( \epsilon ) is used.

3. Corridor (( \gamma_2, \gamma_4 ))

   • As ( \Delta \to 0 ), lines contract; use rectangle to form a closed loop.
   • Cauchy's theorem implies the integral of the closed rectangle is zero.
   • Integral along the small lines vanishes as ( \Delta \to 0 ).

Conclusion

• In the limit ( \Delta \to 0 ), only full circles remain.
• Equation simplified to show integrals of big and small circles are equivalent in the limit.
• Orientation difference accounts for sign change.
• Significant as it allows for substitution with small circles in analysis.
• Assumption: Holomorphic function with a single exception point.

Next Steps

• The result will aid in proving Cauchy's Integral Formula in the next session.