Some Basic Properties of Analytic Functions

Abstract

We identify ℂ with ℝ² by identifying the complex number z = x + iy with the point (x, y) ∈ ℝ². Observe that a (complex-valued) differential form Pdx+Qdy always can be written in the form fdz+gdz̄ where dz = dx+idy and dz̄ = dx-idy (take f = (P-iQ)/2 and g = (P+iQ)/2). This motivates us to introduce the differential operators

$$ \frac{\partial }{{\partial z}} = \frac{1}{2}\left( {\frac{\partial }{{\partial x}} - i\frac{\partial }{{\partial y}}} \right)\quad and\quad \frac{\partial }{{\partial \bar{z}}} = \frac{1}{2}\left( {\frac{\partial }{{\partial x}} + i\frac{\partial }{{\partial y}}} \right) $$

so that

$$ df = \frac{{\partial f}}{{\partial x}}dx + \frac{{\partial f}}{{\partial y}}dy = \frac{{\partial f}}{{\partial z}}dz + \frac{{\partial f}}{{\partial \bar{z}}}d\bar{z} $$

(1.1)

Note that $\Delta = {\partial ^2}/\partial {x^2} + {\partial ^2} = {\partial ^2}/{\partial ^2} = 4{\partial ^2}/{\partial ^2}/\partial z\partial \bar z.$