How to Find the Laurent Series of ez / cos(z) Around z0

In this article, we will walk through the process of finding the Laurent series of the function e^z / cos(z) around z0. We will start by recalling the Taylor series expansions of e^z and cos(z), then proceed with the detailed steps necessary to combine these series.

Step 1: Taylor Series Expansions

The Taylor series for the exponential function and the cosine function are given by:

Exponential Function: e^z sum_{n0}^{infty} frac{z^n}{n!} Cosine Function: cos(z) sum_{n0}^{infty} frac{(-1)^n z^{2n}}{2n!}

Step 2: Explicit Series for cos(z)

We can write out the series for cos(z) explicitly as:

cos(z) 1 - frac{z^2}{2!} frac{z^4}{4!} - frac{z^6}{6!} cdots

Step 3: Finding the Series for 1 / cos(z)

Next, we need to find the series for 1 / cos(z). This can be done using the series expansion for cos(z) and the geometric series formula.

We can express 1 / cos(z) in terms of a power series. Using the series for cos(z):

cos(z) 1 - frac{z^2}{2} - frac{z^4}{24} cdots

We rewrite it as:

frac{1}{cos(z)} frac{1}{1 - left( -frac{z^2}{2} - frac{z^4}{24} cdots right)}

Using the geometric series formula frac{1}{1 - x} sum_{n0}^{infty} x^n where x -frac{z^2}{2} - frac{z^4}{24} cdots.

Step 4: Simplified Series Expansion for 1 / cos(z)

This series can be complicated, but for small values of z, we can approximate:

frac{1}{cos(z)} approx 1 frac{z^2}{2} frac{3z^4}{24} cdots

Step 5: Combining the Series

Now we can multiply the series for e^z and 1 / cos(z):

frac{e^z}{cos(z)} e^z left(1 frac{z^2}{2} cdotsright)

Substituting the series for e^z:

frac{e^z}{cos(z)} left(1 z frac{z^2}{2} frac{z^3}{6} cdotsright) left(1 frac{z^2}{2} cdotsright)

Step 6: Multiplying and Collecting Terms

Multiplying these series and collecting the coefficients for each power of z yields:

1 z z^2 frac{1}{6} z^3 cdots

Conclusion

The Laurent series of e^z / cos(z) around z0 is:

frac{e^z}{cos(z)} 1 z z^2 frac{1}{6} z^3 cdots

This series does not contain any negative powers, indicating that it is actually a Taylor series in this case and is valid in a neighborhood around z0.