A Gentle Introduction to Analytic Continuation

maximios August 15, 2025 Education

Arguably the most important unsolved problem in pure mathematics today is to prove (or disprove) the Riemann hypothesis, which is intimately connected to the distribution of prime numbers. One of the basic techniques needed to understand the problem is called analytic continuation, which is the topic of this article. Analytic continuation is a technique from a branch of mathematics called complex analysis used to extend the domain over which a complex analytic function is defined.

Figure 1: Illustration of the application of the analytic continuation technique to the natural logarithm (imaginary part) (source).

Before introducing the technique, I will briefly explain some important mathematical concepts that will be needed.

Taylor Series

Suppose we want to find a polynomial approximation to some function f(x). A polynomial is a mathematical expression formed by variables and coefficients. They involve the basic operations (addition, subtraction, and multiplication) and contain only non-negative integer exponents of the variables. A polynomial in one variable x of degree n can be written as:

Equation 1: Polynomial in one variable x and degree n. Figure 2: Graphs of polynomials of degree 3 and 4 (source).

Now suppose the polynomial has an infinite degree (it is given by an infinite sum of terms). Such polynomials are called Taylor series (or Taylor expansions). Taylor series are polynomial representations of functions as infinite sums of terms. Each term of the series is evaluated from the values of the derivatives of f(x) at one single point (around which the series is centered). Formally a Taylor series around some number a is given by:

Equation 2: A Taylor series of a function f(x) around a number a.

where the upper indices (0), (1), … indicate the order of the derivative of f(x) as x=a. One can approximate a function using a polynomial with only a finite number of terms of the corresponding Taylor series. Such polynomials are called Taylor polynomials. In the figure below, several Taylor polynomials for the function f(x) = sin x with an increasing number of terms (hence, increasing degrees) are shown.

Figure 3: Taylor polynomials with an increasing number of terms are shown. The black curve is the function sin(x). The other approximations are Taylor polynomials of degree 1, 3, 5, 7, 9, 11, and 13 (source).

The first four Taylor polynomials for f(x) = sin x are given by:

Equation 3: Taylor polynomials for f(x) = sin x with degrees 1, 3, 5 and 7. They are plotted in the figure above (together with higher-order expansions).

Convergence

The concept of convergence of infinite series will also be crucial in our discussion of the analytic continuation. A mathematical sequence is a list of elements (or objects) with a particular order. They can be represented as follows:

Equation 4: Infinite sequence of numbers.

A well-known example of a sequence is the Fibonacci sequence0,1,1,2,3,5,8,13,21,34,55,… where each number is the sum of the two preceding ones.

Figure 4: A tiling with squares that have side lengths equal to successive Fibonacci numbers (source).

One builds a series by taking partial sums of the elements of a sequence. The series of partial sums can be represented by:

Equation 5: Infinite sequence of partial sums.

where:

Equation 6: The values of the partial sums in Eq. 5.

An example of a series, the familiar geometric series, is shown below. In a geometric series, the common ratio between successive elements is constant. For a ratio equal to 1/2 we have:

Equation 7: The geometric series with common ratio = 1/2.

Fig. 5 shows pictorially that the geometric series above converges to twice the area of the largest square.

Figure 5: A pictorial demonstration of the convergence of the geometric series with common ration r=1/2 and first term a=1 (source).

A series such as in Eq. 6 is convergent if the sequence Eq. 5 of partial sums approaches some finite limit. Otherwise, the series is said to be divergent. An example of a convergent series is the geometric series in Eq. 7. An example of a divergent series is:

Equation 8: An example of a divergent series is the so-called harmonic series.

It is easy to see that the harmonic series diverges by comparing it with the integral of the curve y=1/x. See Fig.6. Since the area below the curve is entirely contained within the rectangles and the area below y=1/x is:

Equation 9: The area below the curve y=1/x shown in Fig. 6.

the total area of the rectangles must be infinite as well.

Figure 6: Comparison between the harmonic series and the area below the curve y=1/x proves that the harmonic series diverges.

The geometric series is, in general, the sum of successive powers of a given variable x (see Fig. 5). More concretely, consider the following geometric series where the first term is 1 and the common ratio is x:

Equation 10: An example of a geometric series.