Bessel's Equation

Differential equation bridges the understanding of a complex nth order equation by partitioning each variable and then comprehensively inspecting the relationship between how each variable change with respect to another variable. The unique characteristic about differential equation is its ability to easily associate the equations utilized in mathematic to the technological, scientific and physical world.

Some particular DEs can be modeled as an system of equation which then can be exhibited for its property. The various diverse subjects that relate to differential equation ranges from pharmacy, music, biology, economics, computer science, physics, to all types of engineering such as electrical engineering, mechanical engineering and chemical engineering.

Some would wonder how is it possible that music intertwines with mathematics? Especially differential equations? Well, in the sense that musical instruments releases sound and produces a pattern, differential equation can be used to describe the relationship of the effect of when an acoustic instrument is played. For example, when a drum stick hits a drum pad, it can be analyzed using the Bessel’s equation.

In this paper, we will examine the history of the Bessel’s equation, learn about the background information about the inventor of this equation, determine the purpose of the equation’s creation, and analyze the utilization of the equation from when it was first discovered until now.

Starting off with the discovery of the Bessel differential equation. The Bessel differential equation was founded by Friedrich Wilhelm Bessel, a German astronomer, geodesist, mathematician and physicist, sometime in the 1800s. The exact year the Bessel equation was discovered is questionable since no one knows the true year it was founded.

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Despite that, it is clear that Bessel was the first to systematically proven the set of mathematical functions solution and created a generalized formula of the Bessel function which is universally still used today.

Although Bessel did not reach the standard of some of the most prominent mathematician of his time such as any mathematician from the Bernoulli family’ or someone like Euler, we will dive into the past of Bessel to revisit some background information about his life and his achievements. Friedrich Wilhelm Bessel was born on July 22th, 1784 in a place which at that times was referred to as Minden Westphalia, Minden-Ravensberg, but is nowadays known as Germany. Growing up, Bessel was raised in a poor family. At the young age of 14, Bessel had to work as an apprentice for an import and export business management. Bessel initially started his career as a commercial navigator with the dreams of traveling the world which is why he self-studied geography, languages and navigation. While dealing with business management and being a navigator, he began to apply his mathematical skills to problems he encountered as a navigator. In lieu of his dream of traveling the world, he eventually became an astronomer and mathematician. In 1804, Bessel wrote a paper regarding his observation of the orbit of the Halley’s Comet with the help of data from Harriot and sent it to a German astronomer, Wilhem Olbers, who was so astonished by Bessel’s work that he requested the article to be published instantly which marked Bessel’s first contribution to the astronomy world. This sparked his interest in astronomy and strengthened his desire to study planetary. During this time, Bessel had to choose between his stable management job or take on his first opportunity to work at an observatory. He took the risk and picked the latter; therefore, embarking on his journey of researching and observing the phenomenal of the planets.

One of Bessel’s greatest achievement besides the generalization of the Bessel’s function is his achievement in astronomy. Bessel was the first astronomers who published the systematic approach of calculating the distance from the sun to another star using a method called parallax, which is a trigonometric technique of measuring the distance of an object using the difference in the object’s viewed position from two different sight and the measurement of the angle of the two different sight. At his time, his discovery was recognized as a remarkable discovery because of his precise accuracy in determining the distance to a star. Bessel spent half of his life roughly 28 years to solve this problem. Although Bessel formulated the Bessel equation, through his life’s work it is noticeable that he had a much greater interest in astronomy. He worked at several observatory studying the position of stars. Through his career, Bessel managed to uncover the precise location of approximately 3,222 stars.

Furthermore, another interesting fact about Bessel was Bessel’s invention of the Besselian elements. The Besselian elements is a method to predict the solar eclipses. The method is so accurate that even today NASA still continue to use it to determine when a solar eclipse will occur. A brief overview of how the Besselian element works is that it is a list of data compiled by Bessel that is used to prediction the solar eclipse anywhere on the face of Earth. In fact, although it is uncommon, the Besselian elements has the capability of being applied to stars and planets that orbit the moon. To calculate the solar eclipse, the Besselian element is used to determine the path of two distinct shadows found on the surface of Earth. From what we have learned so far about Bessel, he did not solely work as a mathematician. As he was observing outer space, mathematical calculations had to be done to obtain the data he wanted.

More importantly, while Bessel was researching about one of Kepler’s equation of planetary motion, the motion of three bodies moving under mutual gravitation, he stumbled upon studying what is now known as the Bessel functions. However, through backtracking history, it was later discovered that some of the equations of the set were said to have been already discovered by Daniel Bernoulli, a Swiss mathematician and physicist from the prestigious Bernoulli family, and Leonhard Euler, a Swiss mathematician, astronomer, engineer, physicist and logician who widely influenced analytic number theory and graph theory. The Bessel’s function was surprisingly first discovered and formulated by Daniel Bernoulli in around 1807. According to the book The Britannica Guide to Analysis and Calculus, the author Britannica Education Publishing mentions, “Bernoulli manage to generate equations from the Bessel while investigating the oscillations of a chain suspended by one end. Meanwhile, Euler generated equations while studying the vibrations of a stretched membrane.”

Bessel died on March 17, 1846 in what was then known as Königsberg, Prussia but now known as Kaliningrad, Russia in his observatory from retroperitoneal fibrosis. It is not until after Bessel’s death that the mathematical functions that he studied upon were named Bessel functions. Other than finding the Bessel function, Bessel did not make any other outbreaking discovery in the mathematical industry. During his lifetime, he won the Gold Medal of Royal Astronomical Society twice. Once in 1829 and again in 1841. The Gold Medal of the Royal Astronomical Society is described to be the highest award for an astronomers or geophysicists. The Gold Medal of the Royal Astronomical Society can be awarded for any reasons, but it is generally awarded to those who present outstanding research in either of the field. After his death, to honor his achievements during his lifetime, the largest crater in the Moon’s Mare Serenitatis and a main-belt asteroid was named after him.