Complex Analysis Essentials

Complex analysis studies functions of a complex variable that are differentiable in the complex sense. Its central surprise is that complex differentiability is much stronger than ordinary differentiability of two real variables.

The practical rule is:

Analyticity turns local differentiability into global structure: contour deformation, residues, branch cuts, and pole locations become calculational tools.

Quantum mechanics uses complex analysis in Fourier integrals, propagators, Green functions, scattering amplitudes, resonances, dispersion relations, asymptotic methods, and special functions.

Complex Differentiability

For a function $f:\mathbb C\to\mathbb C$, the complex derivative at $z_0$ is

$$f'(z_0) = \lim_{h\to0} \frac{f(z_0+h)-f(z_0)}{h},$$

where $h$ approaches $0$ through complex values. The limit must be independent of direction in the complex plane.

This is stricter than differentiability as a function of two real variables. For example,

$$f(z)=z^*$$

is not complex differentiable. At $z_0=0$,

$$\frac{f(h)-f(0)}{h} = \frac{h^*}{h}.$$

If $h$ is real, the ratio is $1$. If $h$ is purely imaginary, the ratio is $-1$. The limit depends on direction, so the complex derivative does not exist.

Analytic Functions

A function is analytic, or holomorphic, on a region if it is complex differentiable at every point in that region. For the focused canonical treatment, see Analytic Functions.

Standard examples include:

$$z^n, \qquad e^z, \qquad \sin z, \qquad \cos z,$$

and rational functions away from their poles.

Analytic functions have local power-series expansions. If $f$ is analytic near $z_0$, then

$$f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n$$

inside some disk around $z_0$. This is why analytic continuation can extend a function far beyond the interval where it was first defined.

Cauchy–Riemann Equations

Write

$$f(z) = u(x,y)+iv(x,y), \qquad z=x+iy.$$

For a sufficiently differentiable function, complex differentiability implies the Cauchy–Riemann equations:

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}.$$

These equations express the fact that the derivative is independent of the direction from which $z$ is varied.

For most quantum applications, one does not solve the Cauchy–Riemann equations directly. The key lesson is diagnostic: a function that depends independently on $z$ and $z^*$ is usually not analytic.

Contours

A contour is an oriented path in the complex plane. Complex integration along a contour $C$ is written

$$\int_C f(z)\,dz.$$

Contour methods are powerful because analytic functions can often be deformed without changing an integral, provided no singularities are crossed and the boundary contributions remain controlled.

For the method-level treatment of parametrization, contour deformation, closing Fourier contours, and large-arc estimates, see Contour Integration.

In Fourier-type integrals, one often closes a real-axis integral in the upper or lower half-plane. Which half-plane is allowed depends on exponential decay factors, time ordering, boundary conditions, and the sign of small imaginary prescriptions such as $E\pm i0^+$.

Poles and Residues

A pole is an isolated singularity where a function diverges like a finite negative power of $z-z_0$. A simple pole has the local form

$$f(z) = \frac{a_{-1}}{z-z_0} +\text{analytic terms}.$$

The coefficient $a_{-1}$ is the residue:

$$\operatorname{Res}_{z=z_0} f = a_{-1}.$$

The Residue Theorem says that, for a positively oriented closed contour $C$ enclosing isolated poles,

$$\oint_C f(z)\,dz = 2\pi i \sum_{z_j\ \mathrm{inside}\ C} \operatorname{Res}_{z=z_j} f.$$

This theorem is the engine behind many contour-integral evaluations in Fourier analysis, response theory, and scattering.

Branch Cuts

Some functions, such as $\sqrt z$ and $\log z$, are multivalued unless a branch is chosen. A branch cut is a curve removed from the complex plane so that a single-valued branch can be defined on the remaining region.

Branch cuts appear naturally in quantum mechanics. For a free particle,

$$E = \frac{\hbar^2 k^2}{2\mu}$$

relates energy and momentum through a square root:

$$k(E) = \frac{\sqrt{2\mu E}}{\hbar}.$$

The square root has different sheets. This is one reason scattering amplitudes have branch cuts at thresholds and why resonance poles live on unphysical sheets rather than on the physical real-energy axis.

Analytic Continuation

Analytic continuation extends an analytic function from one region to a larger connected region when the extension is possible. In physics, it is often used to infer hidden structure from values near a physical domain.

Examples include:

• continuing a scattering amplitude away from real positive energy;
• locating bound-state and resonance poles;
• rotating time or frequency variables in controlled settings;
• relating retarded and advanced Green functions through boundary values.

Analytic continuation is powerful but not magic. Singularities, branch cuts, growth conditions, and sheet choices must be tracked.

Why It Appears in Scattering

Scattering amplitudes are functions of energy or momentum. On the physical real axis, they describe measurable scattering. Away from that axis, analytic continuation reveals poles and cuts.

Bound states appear as poles associated with normalizable states. Resonances appear as poles at complex energy, often written schematically as

$$E_{\mathrm{pole}} = E_R-\frac{i}{2}\Gamma.$$

The real part gives a resonance energy scale, while the imaginary part encodes a width. The physics pages Bound States and Scattering Poles and Resonances develop this interpretation.

Why It Appears in Green Functions

A Green function is often a kernel of an inverse operator or resolvent. Formally,

$$G(z) = \frac{1}{z-H}$$

has singularities where $z$ reaches the spectrum of $H$. The small imaginary prescription

$$E\pm i0^+$$

selects different boundary values and therefore different physical conditions, such as outgoing or incoming waves, retarded or advanced response, or causal propagation.

Green Functions gives the operator and boundary-condition framework. The essential complex-analysis idea is already visible here: singularities and boundary values in the complex plane encode spectral and causal information.

The distributional split of $1/(x\pm i0)$ into a principal-value part and a delta part is developed in Principal Value Distributions.

Worked Example: A Simple Residue

Let

$$f(z) = \frac{e^{iz}}{z-i}.$$

The function has a simple pole at $z=i$. Since

$$f(z) = \frac{g(z)}{z-i}, \qquad g(z)=e^{iz},$$

the residue is

$$\operatorname{Res}_{z=i}f = g(i) = e^{-1}.$$

For any positively oriented contour enclosing $i$ and no other singularities,

$$\oint_C \frac{e^{iz}}{z-i}\,dz = 2\pi i\,e^{-1}.$$

Worked Example: Not Every Singularity Is a Pole

The function

$$\log z$$

is not single-valued on the punctured plane. Going once around the origin changes its value by $2\pi i$. The origin is a branch point, not an isolated pole.

This distinction matters in scattering. A pole corresponds to an isolated singularity such as a bound state or resonance. A branch cut often corresponds to a continuum of scattering states or a threshold.

Common Mistakes

• Assuming every complex-valued function is analytic.
• Forgetting that complex differentiability requires direction-independent limits.
• Treating branch cuts as arbitrary decoration rather than bookkeeping for sheets.
• Looking for resonance poles directly on the physical real-energy axis.
• Applying residue formulas when branch cuts or large-arc contributions have not been controlled.
• Ignoring the sign of an $i0^+$ prescription in Fourier or Green-function integrals.
• Moving contours across singularities without adding residue contributions.

Cross-Links

• Complex Numbers
• Analytic Functions
• Contour Integration
• Residue Theorem
• Branch Cuts
• Real Analysis Essentials
• Sequences, Series, and Convergence
• Asymptotic Analysis
• Fourier Transform
• Principal Value Distributions
• Green Functions
• Delta Function
• Scattering Amplitude
• Bound States and Scattering Poles
• Resonances
• Propagator Kernel
• Fourier Transform Table

References

• L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, 1979.
• J. W. Brown and R. V. Churchill, Complex Variables and Applications, 9th ed., McGraw-Hill, 2014.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, 2002.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.

Exercises

1. Show that $f(z)=z^*$ is not complex differentiable at $0$.

Solution

The difference quotient is

$$\frac{f(h)-f(0)}{h} = \frac{h^*}{h}.$$

For real $h$, this equals $1$. For purely imaginary $h=i y$, it equals

$$\frac{-iy}{iy} = -1.$$

The limit depends on the direction of approach, so the complex derivative does not exist.

2. Find the residues of

$$f(z) = \frac{1}{z^2+a^2}, \qquad a>0.$$

Solution

Factor the denominator:

$$z^2+a^2 = (z-ia)(z+ia).$$

The poles are at $z=ia$ and $z=-ia$. The residues are

$$\operatorname{Res}_{z=ia}f = \frac{1}{2ia}, \qquad \operatorname{Res}_{z=-ia}f = -\frac{1}{2ia}.$$

3. Why does $\sqrt z$ require a branch choice?

Solution

Writing $z=re^{i\theta}$ gives

$$\sqrt z = \sqrt r\,e^{i\theta/2}.$$

But $\theta$ and $\theta+2\pi$ describe the same nonzero complex number, while they give square-root values that differ by a sign. To make $\sqrt z$ single-valued, one must choose a branch of the argument, which requires a branch cut.

4. In scattering, why is a pole at $E_R-i\Gamma/2$ not seen as an infinite cross section at real energy $E_R$?

Solution

The pole is at complex energy, generally on an unphysical sheet reached by analytic continuation. Physical measurements use real energies on the physical boundary value of the amplitude. The nearby complex pole produces phase motion and a finite-width line shape rather than an actual divergence on the real axis.