# relief

Suppose I want to get a really good idea about the behaviour of gamma(z) near z=1. What about doing the following:

$\Gamma(\theta_{3}(z,q))\prod_{n=1}^{\infty}\frac{1}{\Gamma\left[(1+2q^{2n-1}\cos(2z)+q^{4n-2}\right]}$

Say this is a relief of the Gamma function in the third Jacobian theta function. Gamma could be replaced with any function whose limit as z approaches 1 is 1. Places where the relief is not one are places where the numerator differs from the denominator, thus taking a very fine microscope to the way the gamma function behaves.

Similarly, we can do the same for infinite sums. All you need is a function which goes to zero as its argument goes to zero. Sometimes you can even do self-reliefs:

$\sin(z) = \sum_{n=0}^{\infty} \frac{(-1)^{n}z^{2n+1}}{(2n+1)!}$

So the self-relief of sine in sine would be:

$\sum_{n=0}^{\infty}\sin\left[ \frac{(-1)^{n}z^{2n+1}}{(2n+1)!}\right] - \frac{(-1)^{n}z^{2n+1}}{(2n+1)!}$