Analytic Functions

An analytic function is a complex-valued function that is complex differentiable throughout an open region. The key point is not the word “differentiable”; it is that complex differentiability is so restrictive that it forces power-series expansions, contour-deformation identities, and strong control over singularities.

This page is the canonical home for analytic and holomorphic functions as mathematical objects. For the method of integrating along deformed complex paths, see Contour Integration. For multivalued functions and sheet choices, see Branch Cuts. For the broader toolkit of poles, residues, branch cuts, and scattering applications, see Complex Analysis Essentials.

Why It Matters

Quantum mechanics uses analytic functions whenever a real variable is promoted to a complex variable to reveal structure:

• Fourier integrals can be evaluated or estimated by moving contours.
• Green functions and resolvents encode spectra through singularities in a complex energy plane.
• Scattering amplitudes use analytic continuation to relate bound states, resonances, and thresholds.
• Special functions used in exactly solvable systems are often defined by analytic differential equations.
• Approximation methods depend on poles, saddle points, branch points, and asymptotic continuation.

The local definition of analyticity is therefore the entrance to a global calculational language.

Complex Derivative

Let $U\subset\mathbb C$ be open and let $f:U\to\mathbb C$. The complex derivative at $z_0\in U$ 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 the direction in the complex plane.

A function is holomorphic on $U$ if this derivative exists at every point of $U$. Many physics texts use “analytic” and “holomorphic” interchangeably. A common convention is:

• holomorphic means complex differentiable on an open set;
• analytic means locally represented by a convergent power series.

For complex functions, these conditions are equivalent on open sets. In this reference, “analytic” usually means either one, unless a local power-series statement is being emphasized.

Cauchy–Riemann Equations

Write

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

where $u$ and $v$ are real-valued functions. If $u$ and $v$ have continuous first partial derivatives, then $f$ is holomorphic exactly when the Cauchy–Riemann equations hold:

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

These equations express direction independence of the difference quotient. Approaching $z_0$ along the real direction gives

$$f'(z_0) = \frac{\partial u}{\partial x} +i\frac{\partial v}{\partial x}.$$

Approaching along the imaginary direction gives

$$f'(z_0) = \frac{\partial v}{\partial y} - i\frac{\partial u}{\partial y}.$$

Equality of the two expressions gives the Cauchy–Riemann equations.

Examples

Polynomials are entire functions, meaning they are analytic on all of $\mathbb C$. For example,

$$f(z)=z^2$$

has derivative

$$f'(z)=2z.$$

The exponential function is also entire:

$$e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}.$$

Its derivative is itself:

$$\frac{d}{dz}e^z=e^z.$$

This is the complex-analysis extension of the phase and oscillation language developed in Complex Exponentials.

Rational functions are analytic wherever the denominator is nonzero. The function

$$f(z)=\frac{1}{z-z_0}$$

is analytic on $\mathbb C\setminus\{z_0\}$ but has a pole at $z_0$.

Nonexamples

The complex conjugation map

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

is not analytic. At $z=0$,

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

For real $h$, this ratio is $1$. For $h=iy$ with real nonzero $y$, it is $-1$. The limit depends on direction, so the complex derivative does not exist.

A function depending separately on $z$ and $z^*$ is usually not analytic. This diagnostic appears often in physics, where expressions may be complex-valued without being analytic functions of a complex variable.

Local Power-Series Structure

If $f$ is analytic in a disk centered at $z_0$, then there is a radius $R>0$ and coefficients $a_n$ such that

$$f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n, \qquad \lvert z-z_0\rvert<R.$$

The nearest obstruction to continuing the power series determines the radius of convergence. That obstruction may be a pole, branch point, essential singularity, or boundary of the domain.

For example,

$$\frac{1}{1-z} = \sum_{n=0}^{\infty}z^n$$

for $\lvert z\rvert\lt1$. The singularity at $z=1$ fixes the radius of convergence around $0$.

This principle is useful in perturbation theory: the convergence of a formal expansion is controlled not only by nearby real values but by singularities in the complex plane.

Consequences of Analyticity

Analyticity has several strong consequences. The following statements are standard, but their hypotheses matter.

Identity theorem. If two analytic functions agree on a set with a limit point inside a connected domain, they agree throughout that connected domain.

Cauchy integral formula. If $f$ is analytic inside and on a positively oriented simple closed contour $C$, then for $z_0$ inside $C$,

$$f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-z_0}\,dz.$$

This formula explains why values inside a region are constrained by boundary values and why analytic functions have derivatives of all orders.

Maximum modulus principle. A nonconstant analytic function cannot have a strict local maximum of $\lvert f\rvert$ inside its domain.

These theorems are not merely decorative. They are the reason analytic continuation is rigid: once an analytic function is known on enough of a domain, much of the rest is forced.

Meromorphic Functions

A meromorphic function is analytic except at isolated poles. Rational functions are the basic examples. Near a pole at $z_0$, a meromorphic function has a Laurent expansion

$$f(z) = \sum_{n=-m}^{\infty} a_n(z-z_0)^n, \qquad m\ge1.$$

The coefficient $a_{-1}$ is the residue. Residues are developed in the broader complex-analysis page and in Residue Theorem; here the important point is that a pole is a controlled failure of analyticity, not an arbitrary divergence.

In quantum mechanics, poles of resolvents or scattering amplitudes often signal discrete spectral features such as bound states or resonances, once the correct physical sheet and boundary conditions are specified.

Analytic Continuation

Analytic continuation extends an analytic function from one region to a larger connected region when the extension is possible. The identity theorem makes this extension unique on the connected domain reached without crossing obstructions.

This is powerful in physics because measured or computed quantities may be given first on a real interval, while their organizing structure lives in the complex plane. Examples include:

• continuing a Green function away from real energy;
• locating poles of a scattering amplitude;
• rotating between real and imaginary time when the hypotheses permit it;
• using special-function identities outside their original real-variable range.

Analytic continuation does not remove singularities. It tells you how to move around them, when that is possible, and when different sheets must be distinguished.

Quantum-Mechanics Uses

For a Hamiltonian $H$, the resolvent

$$R(z) = (z-H)^{-1}$$

is an operator-valued analytic function of $z$ away from the spectrum of $H$. The singularity structure of $R(z)$ encodes spectral data. This is the analytic source of many Green-function prescriptions.

Fourier integrals often contain factors such as

$$e^{ikx}F(k).$$

When $k$ is allowed to be complex, the exponential may decay in one half-plane and grow in the other. Analyticity of $F(k)$ away from isolated singularities then determines whether a contour can be deformed and which singularities contribute.

Scattering amplitudes are usually first interpreted on physical real energies or momenta, but their analytic continuations organize bound states, resonances, and thresholds. The physics interpretation lives in the scattering pages; this page supplies the analytic vocabulary.

Worked Example: Checking Cauchy–Riemann

Let

$$f(z)=z^2.$$

With $z=x+iy$,

$$z^2 = x^2-y^2+2ixy.$$

Thus

$$u(x,y)=x^2-y^2, \qquad v(x,y)=2xy.$$

Compute

$$\frac{\partial u}{\partial x}=2x, \qquad \frac{\partial v}{\partial y}=2x,$$

and

$$\frac{\partial u}{\partial y}=-2y, \qquad - \frac{\partial v}{\partial x}=-2y.$$

The Cauchy–Riemann equations hold everywhere, so $z^2$ is entire.

Common Mistakes

• Calling any complex-valued function analytic.
• Checking differentiability only along the real axis.
• Forgetting that analyticity is defined on open sets, not isolated points alone.
• Treating a branch cut as a failure of the function rather than a choice needed to make a multivalued function single-valued.
• Assuming analytic continuation is unique after crossing a branch point without specifying the sheet.
• Applying pole-based residue reasoning to a branch point or continuum threshold.
• Ignoring growth conditions when using contour deformations in Fourier integrals.

Cross-Links

• Complex Numbers
• Complex Exponentials
• Contour Integration
• Residue Theorem
• Branch Cuts
• Complex Analysis Essentials
• Sequences, Series, and Convergence
• Asymptotic Analysis
• Fourier Transform
• Principal Value Distributions
• Green Functions
• Scattering Amplitude

References

• L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, 1979.
• E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, 2003.
• 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.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.

Exercises

1. Check whether $f(z)=z^3$ satisfies the Cauchy–Riemann equations.

Solution

Write $z=x+iy$. Then

$$z^3 = x^3-3xy^2 +i(3x^2y-y^3).$$

Thus

$$u=x^3-3xy^2, \qquad v=3x^2y-y^3.$$

The derivatives are

$$\frac{\partial u}{\partial x} = 3x^2-3y^2 = \frac{\partial v}{\partial y},$$

and

$$\frac{\partial u}{\partial y} = -6xy = - \frac{\partial v}{\partial x}.$$

So $z^3$ is entire, as expected for a polynomial.

2. Show that $\lvert z\rvert^2$ is not analytic on any open region.

Solution

Since

$$\lvert z\rvert^2 = zz^*,$$

the function depends on both $z$ and $z^*$. More explicitly,

$$f(z)=x^2+y^2,$$

so $u=x^2+y^2$ and $v=0$. The Cauchy–Riemann equations require

$$2x=0, \qquad 2y=0.$$

These hold only at the isolated point $(0,0)$, not on any open region. Hence the function is not analytic on an open region.

3. Find the radius of convergence around $z=0$ for

$$f(z) = \frac{1}{1+z^2}.$$

Solution

The singularities occur where

$$1+z^2=0,$$

so $z=i$ and $z=-i$. Both are distance $1$ from the origin. Therefore the Taylor series about $0$ has radius of convergence $R=1$.

4. Why is analyticity relevant to a Fourier integral such as $\int e^{ikx}F(k)\,dk$?

Solution

If $k$ is complex, the factor $e^{ikx}$ can decay in one half-plane and grow in the other, depending on the sign of $x$. If $F(k)$ is analytic in the region through which the contour is moved, the contour can often be deformed without changing the integral. Singularities crossed during deformation give additional contributions, which is how poles and branch cuts enter many Green-function and scattering calculations.