Magnification in complex analysis refers to the phenomenon that occurs when a function is applied to a small disc in the complex plane.
The first step to understanding this concept is to understand the idea of a complex plane. The complex plane is a geometric representation of complex numbers. A complex number is a number that can be written in the form a + bi, where a and b are real numbers, and i is the square root of -1.
When a function of a complex variable is applied to a small disc in the complex plane, the disc may undergo several transformations. It could potentially be expanded or contracted by the function. This process of alteration gives rise to what is termed as "magnification."
To assess the magnification, one needs to examine the ratio of the magnitude of the original disc radius to the magnitude of the transformed disc radius.
The higher the ratio, the greater the level of magnification. This indicates that the function has blown up the small disc to a larger extent. Conversely, a lower ratio signifies that the function has contracted the disc. In both situations, the magnitude of the transformation is captured quantitively by this ratio. This is the concept of magnification in the realm of complex analysis.