Complex Analysis and Analytic Continuation

Why can a contour integral determine a physical calculation, and when does moving the contour change the answer? This chapter connects holomorphy, singularities, analytic continuation, and boundary values. It supplies the mathematical conditions for contour arguments; the physical origins of analyticity require separate justification.

Enter this chapter

Begin with Holomorphic Functions and Cauchy Theory to understand which contour deformations preserve an integral. Readers handling propagator poles can then proceed to residues and causal prescriptions.

Topics

1. Holomorphic Functions and Cauchy Theory — Derives the relation between holomorphy, contour integrals, and local derivatives.
2. Laurent Series, Poles, and Residues — Extracts contour integrals from isolated singularities and winding numbers.
3. Branches, Sheets, Analytic Continuation, and Monodromy — Follows a chosen analytic branch along paths and around singularities.
4. Boundary Values, Discontinuities, and Dispersion Integrals — Relates cut discontinuities to analytic functions with the necessary subtraction data.
5. Contour Deformation, Pinches, and Causal Prescriptions — Diagnoses crossed poles, branch cuts, large arcs, and pinched contours.

How the ideas fit together

Residues and dispersion integrals are two forms of the same contour argument. Isolated poles contribute local Laurent coefficients; a cut contributes the difference between its two boundary values. Neither calculation determines a function from those singularity data alone when a contour at infinity survives: subtraction constants or polynomials can remain. The distinction between singularity data and asymptotic assumptions is developed in Zwicky 2016, § 2.2.

A branch cut is a choice for displaying continuation, whereas a pinch obstructs the proposed deformation itself. A Wick rotation therefore requires control of poles, cuts, endpoints, and infinity; substituting an imaginary time variable is not by itself a proof.

Review the chapter

Can two functions have identical discontinuities across a cut and still differ?

Solution

Yes. Adding a polynomial changes no discontinuity because a polynomial is entire. Growth conditions bound which additions are allowed, and subtraction values fix the remaining freedom. Discontinuity data without those conditions are incomplete.

Where to go next

Continue to Unitarity and Analyticity to connect these mathematical tools with physical intermediate states and amplitude singularities.

References

• Roman Zwicky, “A Brief Introduction to Dispersion Relations and Analyticity,” lectures at the Dubna School “Strong Fields and Heavy Quarks” (2016), arXiv:1610.06090. Stable preprint.