Carl Jacobi (1809 - 1851)

Jacobi was one of three children born into a wealthy German banking family at the beginning of the nineteenth century. Jacobi gained a good early education which enabled him to enter the Gymnasium (what we would call a secondary school) in Potsdam. where almost immediately his natural abilities were noticed and he was put into the final year class - not bad for his first year at school!

This meant that by the end of the academic year he was good enough to enter the University of Berlin. However, as the young Jacobi was only 12, he was not accepted. He spent his enforced waiting time - until he was 16 - studying alone, becoming very competent in Latin, Greek, and history and, of course, mathematics.

He was eventually admitted to the University of Berlin and chose mathematics as his speciality. However, the standard of mathematics education wasn’t great so he continued his school habits and taught himself .He soon won the right to teach secondary school pupils in all his own favourite subjects and joined the teaching staff at one of the best schools in Berlin.

Jacobi’s first paper, submitted to the Academy of Sciences in Berlin in 1825, was not a success. It dealt with some important aspects of symmetries in special functions and was not published during his lifetime.

This was not a good start but he soon began a very creative and productive time that saw him working on a number of problems and producing papers of his results - usually making fundamental advances.

In order to help his career prospects Jacobi changed his religion and become a Christian. At the time it was almost impossible for a non-christian to obtain employment and recognition in a university.

In September 1831 Jacobi married Marie Schwinck. A few months later in May 1832, he was promoted to full professor in the university of Berlin - a university in which he spent most of his professional life. Jacobi’s reputation as an excellent teacher attracted many students and his own research and the esteem in which it was internationally held made Berlin a leading mathematical institution.

In 1843 Jacobi was diagnosed with diabetes and was advised to move to a warmer climate to recuperate. Unfortunately, Jacobi was not a wealthy man - a European wide depression had left him bankrupt. After some efforts by the mathematical community, a grant was obtained from Friedrich Wilhelm IV. With this in place, Jacobi set off for Rome in November 1843. Though his health improved, it was decided that his last teaching post - at Königsberg - was not appropriate so Jacobi transferred back to Berlin, again with special financial assistance.

Civil unrest followed due to unemployment and crop failures which led to uprisings across Germany, including Berlin. Jacobi had managed to offend both sides in the argument so some of his earlier financial help was withdrawn and he was forced to move to a small town - Gotha - with his family. A few months later his acceptance of a chair at the University of Vienna spurred the German government to offer a strange compromise. They provided the finance for him to work in Berlin, but not sufficient that his family could join him. They stayed in Gotha. In January 1851 he contracted influenza, swiftly followed by smallpox which killed him within days.

Jacobi’s Mathematics

Jacobi was a master of understanding functions made up of an infinite number of terms all multiplied together - their product. For example, consider the function f where,

f(x)=(1+x)(1+qx)(1+q^2x)(1+q^3x)\cdots

What, Jacobi wondered, happens if this is expanded by actually multiplying out the brackets ? Well, of course, that is not possible, since there are an infinite number of terms. Nonetheless, Jacobi got round that profound difficulty. Here’s how. He started by moving the first term of the product to the other side of the equation:

{f(x)\over{(1+x)}}=(1+qx)(1+q^2x)(1+q^3x)\cdots

and then Jacobi’s master stroke - he noticed that what remained on the right was the function itself, slightly altered,

{f(x)\over{(1+x)}}=(1+(qx))(1+q(qx)(1+q^2(qx))\cdots=f(qx)

So he wrote this in the form,

f(x)=(1+x)f(qx)

and this is a ‘symmetry’ of the function. Now Jacobi considered what might happen if the original product were multiplied out:

f(x)=1+a_1x+a_2x^2+a_3x^3+\cdots

in which the terms a involve q. But what might they be ? Jacobi popped them into his symmetry,

f(x)=1+a_1x+a_2x^2+a_3x^3+\cdots=(1+x)f(qx)=(1+x)\left(1+a_1qx+a_2q^2x^2+a_3q^3x^3+\cdots\right)

and then multiplied out the terms on the right:

1+a_1z+a_2z^2+a_3z^3+\cdots=1+a_1qz+a_2q^2z^2+a_3q^3z^3+\cdots+z+a_1qz^2+a_2q^2z^3+a_3q^3z^4+\cdots

\qquad\qquad\qquad\qquad\qquad\ \ =1+(a_1q+1)z+(a_2q^2+a_1q)z^2+(a_3q^3+a_2q^2)z^3+\cdots

But the terms in x, in z^2, in z^3,\cdots must be the same on both sides, and if they are, then:

a_1=a_1q+1,

a_2=a_2q^2+a_1q,

a_3=a_3q^3+a_2q^2

If the first equation is solved - for a_1 - then the second can be solved for a_2 and then the third for a_3\cdots. Jacobi had cracked the problem:

a_1=a_1q+1\mbox{, and so,}\qquad a_1(1-q)=1

\Rightarrow \qquad\qquad\qquad\qquad\qquad\qquad a_1={1\over{1-q}}

a_2=a_2q^2+a_1q\mbox{, and now, }a_2(1-q^2)=a_1q={q\over{1-q}}

\Rightarrow \qquad\qquad\qquad\qquad\qquad\qquad a_2={q\over{(1-q)(1-q^2)}}

a_3=a_3q^3+a_2q^2\mbox{, so that, }a_3(1-q^3)=a_2q^2={q.q^2\over{(1-q)(1-q^2)}}

\Rightarrow \qquad\qquad\qquad\qquad\qquad\qquad a_3={q^3\over{(1-q)(1-q^2)(1-q^3)}}

Can you see the pattern ? Jacobi wrote it like this:

a_n={{q^{n(n+1)\over2}}\over{(1-q)(1-q^2)\cdots(1-q^n)}}

In Jacobi’s hands, such ideas could be applied to many problems, especially in number theory.

Another problem that Jacobi made fundamental advances in was the theory of elliptic functions. This started as the problem of finding the length of certain curves - the perimeter of an ellipse, gives the subject its name:

The functions that give the length of curves such as this all have a number of fundamental symmetries - much as the infinite product above has. They rely on results in calculus - involving rather complicated integrals. The integrals themselves cannot be evaluated exactly, but their behaviour - as a function - gives rise to a new subject, of greater importance than the starting problem of finding the length of curves. The functions and especially their inverses generate a new subject that resembles trigonometry - which is based on the circle - but is much more complicated and subtle. The results it provides, however, give insights and powerful ideas that can be applied in many different areas of mathematics. Jacobi built on work that Gauss had started, Legendre would continue and Abel would transform.

Jacobi also carried out important research in another branch of calculus - partial differential equations. These equations became important as a result of the work of scientists - especially physicists. They arise in many areas of physics - the study of heat, dynamics and later electromagnetism. The equations he studied are still not fully solved but the number of applications continues to grow - the flow of heat provides a model for the ‘flow’ of information and knowledge. Understanding this and predicting its behaviour is important in the Stock market and banking. Dynamics provides the means to plan the paths of space probes, and even the motion of planets and larger systems such as galaxies. These too have symmetries - a planet in orbit about a sun, returns to the same place on a regular basis. Electromagnetism is the means by which the internet, the computer, TV, radio and all the other modern forms of communication operate.

Jacobi lent his name to many subjects, so that today we still speak of the Jacobian, the Jacobian triple identity, and of the Jacobi Theeta functions. Few mathematicians have been so honoured.