The following examples illustrate how different analytic functions defined on a punctured disk can be applied in quantum mechanics (QM) and quantum field theory (QFT):

1. Single-Pole Functions

f(z) = 1 / (z - z₀)

Use Case: This form appears in Green’s functions or propagators. For example, in QM, the Green’s function for a 1D free particle is:

G(E) = 1 / (E - E₀)

In QFT, propagators for particles often have poles at the particle’s mass m in momentum space:

Δ(p²) = 1 / (p² - m²)

2. Higher-Order Poles

f(z) = 1 / (z - z₀)ⁿ, n ≥ 2

Use Case: Higher-order poles appear in renormalization or when studying higher derivatives of Green’s functions or scattering amplitudes. Residues at such poles provide information about subleading corrections in perturbation theory.

3. Laurent Series

f(z) = Σ aₙ (z - z₀)ⁿ, where aₙ ≠ 0 for some n < 0

Use Case: Laurent series expansions are used in contour integration techniques in QFT, particularly in the calculation of loop integrals. The coefficients aₙ for n < 0 represent contributions from singularities (poles), crucial for defining scattering amplitudes via the residue theorem.

4. Logarithmic Functions

f(z) = ln(z - z₀)

Use Case: Logarithms frequently arise in quantum corrections. For example:

• In renormalization group equations, terms like ln(μ), where μ is a renormalization scale, describe how coupling constants evolve with energy.
• In QM, phase shifts in scattering often involve logarithmic terms due to boundary conditions or potential wells.

5. Exponentials and Oscillatory Functions

f(z) = exp(1 / (z - z₀))

Use Case: Exponentials of this type are seen in semiclassical approximations, like the WKB method:

ψ(x) ~ exp(iS(x) / ħ)

Such forms are common in tunneling problems, where S(x) may have singularities.

6. Meromorphic Functions

f(z) = sin(z - z₀) / (z - z₀)

Use Case: Meromorphic functions arise in spectral analysis of quantum systems. For instance, sin(z) / z is related to spherical Bessel functions, which describe the radial part of wavefunctions in quantum scattering problems.

7. Fractional Power Functions

f(z) = (z - z₀)^(1/2)

Use Case: Fractional power functions appear in branch cuts associated with multi-valued quantities, such as the square root of momentum in potential scattering. They also arise in the study of Riemann surfaces used in QFT for complex-valued momenta.

8. Rational Functions Excluding the Puncture

f(z) = (z² + 1) / (z - z₀)

Use Case: Rational functions describe propagators and resonances in QFT. For example, the Breit-Wigner resonance is rational:

G(p) = 1 / (p² - m² + iε)

Such forms model the decay of unstable particles.

General Connection

These functions are widely used in:

• Scattering Theory: To describe wavefunctions, scattering amplitudes, or S-matrix elements in QM or QFT.
• Complex Analysis in QFT: Analytic continuation and residue calculations often involve these functions.
• Path Integrals: Singularities in propagators or effective actions often include these types of functions.

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