# Analytic Function

The theory of complex functions mainly deal with analytical functions.

$$f'(z) = \frac{d}{dz}f(z) = \lim_{\Delta z\to 0}\frac{f(z+\Delta z) - f(z)}{(z+\Delta z) - z}= \lim_{\Delta z\to 0}\frac{f(z+\Delta z) - f(z)}{\Delta z}$$

Let $$\Delta z = \Delta x$$. $$f'(z) = \lim_{\Delta x\to 0}\frac{f(z+\Delta x) - f(z)}{\Delta x} = \lim_{\Delta x\to 0}\frac{u(x+\Delta x,y) + iv(x+\Delta x,y) - u(x,y) -iv(x,y)}{\Delta x} = \lim_{\Delta x\to 0}\frac{[u(x+\Delta x,y) - u(x,y)] + i[v(x+\Delta x,y) -v(x,y)]}{\Delta x} = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x}$$

Let $$\Delta z = i\Delta y$$. $$f'(z) = \lim_{\Delta y\to 0}\frac{f(z+i\Delta y) - f(z)}{i\Delta y} = \lim_{\Delta y\to 0}\frac{u(x,y+\Delta y) + iv(x,y+\Delta y) - u(x,y) -iv(x,y)}{i\Delta y} = \lim_{\Delta y\to 0}\frac{[u(x,y+\Delta y) - u(x,y)] + i[v(x,y+\Delta y) -v(x,y)]}{i\Delta y} = -i\frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}$$

$$\frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} = -i\frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}$$.

## Definition

A function $$f$$ is analytic if $$f$$ has a derivative at $$z$$ and in the neighborhood of $$z$$. (I.e. in a region $$\mathcal{R}\subset\mathbb{C}$$)

Also called entire or holomorphic.

Entire means that $$f$$ is holomorphic in $$\mathcal{R}=\mathbb{C}$$.

Meromorphic means that $$f$$ is holomorphic in $$\mathcal{R}=\mathbb{C}$$ except for a set of isolated poles.

### Cauchy-Riemann Conditions

Thus, $$\frac{\partial u}{\partial x} = i\frac{\partial v}{\partial y}$$ and $$-\frac{\partial v}{\partial x} = \frac{\partial u}{\partial y}$$ This is called the Cauchy-Riemann Conditions.

### Aside

If you can write a function $$f$$ solely in terms of $$z$$ (not $$z^*$$) (e.g. $$f(z) = 1/z,\sin z,z^2$$) then it is differentiable everywhere it is finite. An example of one that is not, $$|z|^2 = zz^2$$ is not entire.

If it is differentiable then all of the derivative rules from before hold. I.e. $$\frac{d}{dz}z^n = nz^{n-1}$$ etc.

Created: 2023-06-25 Sun 02:24

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