After 10 years of research, Professor Raimar Wulkenhaar of the Institute of Mathematics at the University of Münster, and Dr. Erik Panzer of Oxford University, solved a mathematical equation that is considered unsolvable. This equation is used to solve the problems raised by elementary particle physics. In this article, we will follow Professor Wulkenhaar to review the challenges and surprises in the search for solutions.
 

This is a nonlinear integral equation with two variables. Such an equation is very complicated and it is hard to believe that there will be any formula as a solution to the problem. The two variables alone are a challenge, not to mention that mathematicians do not have a clear way to find solutions to nonlinear integral equations.
 

However, in the past decade, the light of hope has flashed again and again, and finally Wulkenhaar and collaborators have ruled out all the difficulties, and finally convinced that finding a clear solution to the problem (expressed by known functions) is actually possible.
 

The equation expresses a mathematical understanding of quantum field theory and belongs to the physical field and can be applied to large-scale experiments, such as particle collision experiments conducted by CERN. The goal is to mathematically describe elementary particles. However, the problem is so complex that things have to be reversed, that is, mathematically describing the particles of the idea using the deterministic properties of the actual particles. It is hoped that one day, actual particles can be described using methods established in this way.
 

At the end of May, Wulkenhaar tried a new idea, the decisive inspiration for this idea came from his PhD student Alexander Hock. He solved a new equation, which was simpler than the previous equation, and then began to solve layer by layer. This means that, step by step, loop iteration, the left side of the equation calculated in the previous step is substituted into the right side of the equation in the next step.
 

In the fourth layer loop iteration, you need to calculate the sum of 46 integrals, including the polylogarithm. These multiple logarithmic functions become more complex in each iteration. Fortunately, in the final summation, almost all of the parts are offset, and the rest is only the sum of the exponents of the short ordinary logarithmic function. He immediately realized that there was a treasure in it.
 

○ Multiple logarithmic function is a special power series, which can also be expressed as its own integral, forming an iterative structure. In quantum mechanics, the multiple logarithmic function appears as a closed form of the integral of the Fermi-Dirac distribution and the Bose-Einstein distribution. The figure shows several different multiple logarithmic functions on a complex plane. | Image source: Wikipedia.
 

The fifth layer of iterative solution is not so easy to solve, but this time luck is still with him. In a summer school in the French Alps, Wulkenhaar had the opportunity to talk to experts who studied these equations, one of whom was Erik Panzer of Oxford University. He has written and supported computer programs for hyperlogarithm in symbolic mathematics.
 

Overnight, the program went to the seventh level of circulation. The computer program verifies the result of the Wulkenhaar solution to the fourth layer of the loop, after which the miracle continues and everything is broken down into ordinary logarithmic functions. A pattern has appeared!
 

Perhaps you remember the Pascal's Triangle as a binomial coefficient learned in the student days? In the Pascal triangle, each number is the sum of the two numbers above it.
 

What Wulkenhaar found in the loop is a triangular structure like this, perhaps just a little more complicated than the Pascal triangle.
 

On June 9, the 8th and 9th layers were completed. Then, the most important moment has arrived. Erik Panzer cracks a so-called recursive formula that produces the next row of numbers from the top row of the triangle, and they are thus inferred from the unknown.
 

At this time, Wulkenhaar realized that they would solve this problem. He is fortunate to say: "No one will be so lucky."
 

The next day, Wulkenhaar succeeded in simplifying a part of the equation into a simple derivative series. Initially, the rest of the problem seemed difficult. In the middle of the night, he suddenly thought of solving the problem with the Cauchy formula. This attempt proved to be correct. In the next step, he used a formula that he often saw. He realized that it could be solved with the Lambert W function.
 

○ The Cauchy integral formula is an important conclusion of complex analysis, named after the mathematician Augustin-Louis Cauchy (1789-1857). The Cauchy integral formula shows that for a holomorphic function on any closed region, the integral value inside the region depends entirely on the value at the region boundary.
 

A few minutes later, Erik Panzer's mail came: he also thought of the Lambert function, but through a completely different path. As a result, they solved the problem that seemed unsolvable in 10 years—the solution to the integral equation describing the quantum field theory. Wulkenhaar lamented: "This is incredible!"
 

The Lambert function is named after the Swiss mathematician Johann Heinrich Lambert (1728-1777). This equation appears in many different problems, such as the Fermi-Dirac distribution and the Bose-Einstein distribution. Because of the lack of understanding of Lambert's basic work, in history, the Lambertian function was "invented" over and over again until 1993, when the Lambertian function was established as the standard.
 

Wulkenhaar used a lot of 18th century development methods in the solution process, and they were almost completely forgotten in the dust of history. It is these old formulas that helped him.
 

Wulkenhaar also solves an integral problem with the Lagrange-Bürmann formula, along with the Lambertian function and the Cauchy formula. In general, the field of mathematics gives great respect to the predecessors. The names Euler, Lambert, Lagrange, Cauchy, Gauss, and Hilbert are remembered in the formula, expressing the highest recognition of their achievements.
 

However, there are two modern tools that have helped Wulkenhaar: Wikipedia and computers. He can use Wikipedia to retrieve well-known, or little-known, information about mathematical structures and equations. The computer can solve the equation at a speed that is unmatched by hand and does not make any mistakes.
 

A new function appears in the solution of the equation, named Nielsen function. When everything is better understood, such as how the Nielsen function is related to other functions, they will submit their own research results. Later, Wulkenhaar will continue to study a problem that has been studied since 2002. This issue is also related to quantum field theory.