Here is my latest announcement to my first-year students.

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In the spirit of “applications of pure mathematics”, I thought I would say something about applications of complex numbers.

According to the Wikipedia page

https://en.wikipedia.org/wiki/Complex_number

complex numbers were first introduced by an Italian mathematician, Gerolamo Cardano, during his attempts to solve cubic equations in the 16th century.

You probably all know the quadratic formula. There are similar but more complicated formulae for solving cubic and quartic polynomials. The search for a similar formula for the quintic proved fruitless, and in fact there is, in general, no such formula for solving the quintic. The relevant area of mathematics is Galois Theory. This is off-topic today, but see https://en.wikipedia.org/wiki/Galois_theory if you want a flavour. This is not to say that quintics don’t have roots (they do!), just that you can’t always find a formula for them using the coefficients and nth roots etc.

This is all well and good, but inventing some apparently fictitious numbers in order to find solutions where you didn’t have them before may not feel like much progress. But the amazing thing is that “pure” theory of complex numbers, complex functions and complex analysis has applications almost everywhere you look, and not just within mathematics.

If you have studied physics, you may already have met complex numbers and functions when looking at impedance, phase angles, and oscillating currents. Wikipedia mentions practical applications in many other fields. I’m only going to mention a small number of things today, but you could look at

https://en.wikipedia.org/wiki/Complex_number#Applications

for more.

In first year calculus, when you study differential equations, you will see some complex numbers come in when looking for solutions. They then go away again, because you want to find solutions using real numbers. But the exponentials of imaginary numbers lead you to use the functions cos and sin in your solutions.

In second-year complex functions you will see how the beautiful theory of complex functions enables you to use “residue calculus” to quickly find the exact values of “improper integrals” that look a little tricky otherwise, such as

and many far more complicated examples. In fact this topic is enough on its own for an third-year project! But you could see

https://en.wikipedia.org/wiki/Methods_of_contour_integration#Applications_of_integral_theorems

for a few more examples.

I think that it is remarkable that the most efficient way to calculate this kind of real integral involves using the theory of complex functions as (mostly) developed in the 19th century, especially the work of Cauchy and Riemann.

I could say much more here, but for now I’ll just mention that these methods become crucial again for calculating the Laplace transform and inverse Laplace transform, which have too many applications to list here! See, for example,

https://en.wikipedia.org/wiki/Laplace_transform

Best wishes,

Dr Feinstein