What is e?
The constant e is a transcendental number that can be obtained as:
In general, the exponential function plays an important role in processes where the change of a quantity is related to its magnitude (processes usually described by differential equations).
The value of e:
But better look here: http://numbers.computation.free.fr/Constants/E/e.html
Importance in crystallography
Periodically varying functions can be expressed either as sines and cosines, or as complex exponentials. eix is periodic, and can be used to describes X-ray waves (or periodic/oscillating physical processes in general).
The Euler's formula:
is the basic relationship between these two representations. If we expand each of the functions above in an infinite series, we can prove that: