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Complex Analysis
Explore complex functions; integration and singularities; mappings and global behavior. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Derivatives
A derivative is the limit of a difference quotient, measuring local rate of change when that limit exists.
7 reviewed resources
Complex functions
Complex numbers
A complex number has real and imaginary parts and is represented on the complex plane.
Video explanations to be addedComplex exponentials
Euler's formula links complex exponentials to cosine and sine for real angles.
Video explanations to be addedHolomorphic functions
A function is holomorphic when complex differentiable throughout an open neighborhood.
Video explanations to be addedCauchy–Riemann equations
Complex differentiability imposes relations between real and imaginary partial derivatives.
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Integration and singularities
Complex contour integrals
Complex integration accumulates function values along an oriented parameterized curve.
Video explanations to be addedCauchy integral formula
Contour values determine interior values and derivatives of a holomorphic function under the contour hypotheses.
Video explanations to be addedLaurent series
A Laurent series uses positive and negative powers on an annulus.
Video explanations to be addedIsolated singularities
A punctured neighborhood can contain a removable singularity, a pole or an essential singularity.
Video explanations to be addedResidues
The coefficient of power minus one in a Laurent expansion controls a local contour integral.
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Mappings and global behavior
Residue theorem
A contour integral equals two pi i times enclosed residues under the theorem's hypotheses.
Video explanations to be addedConformal maps
A holomorphic map with nonzero derivative locally preserves oriented angles.
Video explanations to be addedAnalytic continuation
Compatible holomorphic functions extend an analytic function to larger connected regions.
Video explanations to be added