How to visualise Laurent Series Coefficients?

Question

I'm trying to think about this same as Taylor series $\displaystyle{f(z) = \sum_{k = 0}^{\infty} a_k\,(z-z_0)^k}$, wit coefficients $a_k = \dfrac{f^{k}(z_0)}{k!}$. You can easily imagine how $f(z)$ is approximated at $z_0$ by adding more and more polynomial parts of higher dimension (First it's a line, then a parabola etc.)

I have problems to extend this idea on Laurent series, learning about this right now: $f(z) = \displaystyle{f(z) = \sum_{k = 0}^{\infty} a_k\,(z-z_0)^k} + \sum_{k = 0}^{\infty} b_j(z-z_0)^{-j}$ with $a_k = \dfrac{1}{2\,\pi\,i} \int_C\dfrac{f(z)}{(z-z_0)^{k+1}}\,\mathrm{dz}$ and $b_j = \dfrac{1}{2\,\pi\,i} \int_C\dfrac{f(z)}{(z-z_0)^{-n+1}}\,\mathrm{dz}$.

I do know Laurent series is used when the point $z_0$ where you build your series from is not analytic expressing it with the help of coefficients $b_j$. Also Laurent Series just seems to work in a domain $n_1 < \vert z\vert <n_2.$ This might make it hard to visualise the whole process. The best I can do is thinking about $f(z)$ as a 2 Dimensional function (magnitude of $f(z)$ over the imaginary-real-plane) approximated by a 2 Dimensional Taylor Series. Just that not analytic points $z_0$ like singularities are included. This does not explain why the series is restricted to $n_1<\vert z \vert <n_2$ where it only converges.

Answer 1

One way to visualise the meaning of Laurent series is to use the Fourier series.

Let us look at the values of function $f(z)$ along the circle given by $|z-z_0|=r$, where we asume that this circle lies in the region of the convergence of the given Laurent series. If $$ f(z) = \sum_{k=-\infty}^\infty a_k (z-z_0)^k $$ then $$ f(z_0+re^{i\phi}) = \sum_{k=-\infty}^\infty a_k r^k e^{ik\phi} $$ That means that if $a_k$ are the coefficients of the Laurent series of function $f(z)$, then $a_kr^k$ are the coefficients of the Fourier series (in the exponential form) of the function $g(\phi) = f(z_0+re^{i\phi})$. And just as Fourier series is the decomposition of a periodic function into trigonometric functions, the Laurent series is a decomposition of a holomorphic functions into function that "oscilate" around point $z_0$: function $(z-z_0)^0 = 1$ has no oscilations, function $(z-z_0)$ has one oscilation around the circle, function $(z-z_0)^k$ has $k$ oscilations, function $(z-z_0)^{-k}$ also has $k$ oscilations, but in a sense "in the oposite direction".

As for the the understanding of the convergence of the Laurent series, the division $$ f(z) = \sum_{k=0}^\infty a_k(z-z_0)^k + \sum_{k=1}^{\infty} a_{-k} (z-z_0)^{-k}$$ Notice that if $|z-z_0|$ is too big, then $|z-z_0|^{k}$ may grow too fast for the series $\sum_{k=0}^\infty a_k(z-z_0)^k$ to converge; on the other hand, if $|z-z_0|$ is to small, then $|z-z_0|^{-k}$ may not go down fast enough for the series $ \sum_{k=1}^{\infty} a_{-k} (z-z_0)^{-k}$ to converge. If both too big and too small values of $|z-z_0|$ are not allowed, you get a condition of the form $n_1 < |z-z_0| < n_2$.