This week we celebrate the birthday of Augustin-Louis Cauchy, a French mathematician, physicist and engineer whose work is recognised by maths students everywhere (and by students of physics and engineering too) thanks to the countless theorems, lemmas and properties that have been given his name.

(Check out the full list on Wikipedia, it really is a lot https://en.wikipedia.org/wiki/List_of_things_named_after_Augustin-Louis_Cauchy).

To mark the occasion here are three of the best:

Cauchy’s Integral Theorem

Most recognised by the equation

∮_γ▒〖 f(z) dz〗  = 0

this complex analysis theorem states that for any path γ that connects two given points, then assuming a function f has certain properties, the corresponding path integrals of the function will be the same. Its applications are varied but perhaps most interestingly in proving and deriving several other concepts such as Louisville’s Theorem (that represents densities of particles in 6 dimensions), the fundamental theorem of algebra (which defines properties of the complex numbers), and the appropriately named Cauchy’s estimate.

Cauchy’s Root Test (Or Cauchy’s radial test)

We learn about sequences in mathematics from a very young age, spotting that after 2 ,4, 6 we expect to see 8 or as we get older that after 1/2,1/4,1/8  we know 1/16 will follow. If we were to add up each term of a sequence indefinitely, we would then instead have a series. Cauchy’s Root test is used to demonstrate a property of a series known as convergence. (It is carefully defined in mathematics but broadly speaking represents, as you would expect, that as more and more terms of the sequence are added together, the total will become closer and closer to ‘converging’ on some point.)

The root test uses the quantity

C = lim┬(n→∞)⁡〖sup〖√(n&|a_n | )〗^ 〗

To demonstrate if the series satisfies this property of convergence, depending on whether C is less than, equal to or greater than 1.

Cauchy’s Crater

Not technically related to his work however this 120km in diameter crater on the surface of the moon has too been named after the famous mathematician, demonstrating how highly regarded his research and achievements are in the science.

Maths in (social) media…

Group theory is no joke.

The beauty of primes.