Complex analysis

Complex analysis is, broadly speaking, the study of functions that take as input complex numbers and output complex numbers, and behave well with respect to a notion of complex differentiation, as discussed below. It's crucial to note that just having complex-valued functions does not qualify something for being called complex analysis; it is really the new definitions of differentiation and integration with respect to complex variables, and using the field structure of complex numbers, that makes the subject different.

Complex analysis is closely related to real analysis, the study of functions over the reals. However, a number of beautiful results that hold in complex analysis, fail to have analogues in real analysis. The core reason behind this is that the complex numbers form a plane, far more well-connected than the real line (which can be disconnected by just removing one point) allowing for fascinating arguments with geometric content.

Differentiation
Let us now turn to the question: Is it possible to extend the methods of calculus to functions of a complex variable, and why might we want to do so? We recall the definition of one of the two fundamental operations of calculus, differentiation. Given a function $$y = f(x)$$, we say f is differentiable at $$x_0$$ if the limit


 * $$\lim_{h\to 0} \frac{f(x_0 + h) - f(x_0)}{h}$$

exists, and we call the limiting value the derivative of f at $$x_0$$, and the function that assigns to each point x the derivative of f at x is called the derivative of f, and is written $$f'(x)$$ or $$df/dx$$. Now, does this definition work for functions of a complex variable? The answser is yes, and to see why, we fix x and unravel the definition of limit. If the limit exists, say $$c = f'(x)$$, then for every (real) number $$\varepsilon > 0$$, there is a (real) number $$\delta > 0$$ such that if $$|h| < \delta$$ then


 * $$\left | \frac{f(x + h) - f(x)}{h} - c \right | < \varepsilon$$

This makes perfect sense for functions of a complex variable, but we need to keep in mind that $$| \cdot |$$ represents the modulus of a complex number, not the real absolute value.

This seemingly innocuous difference actually has far reaching implications. Recall that the complex plane has two real dimensions, so there are many ways that h can approach 0: successive values of h may be points on the x-axis, points on the y-axis, some other line through the origin, it may spiral in, or take any of a number of paths, but the definition requires that the limit be the same number in every case. This is a very strong requirement! Fortunately, it turns out to be sufficient to consider just two of the possible "approach paths": a sequence of values along the x-axis and a sequence of values along the y-axis. If we call the real and imaginary parts (respectively) of $$w = f(z)$$ u and v, (i.e., $$w = f(z) = u + iv$$), this requirement can be expressed in terms of the partial derivatives of u and v with respect to x and y (assuming existence and continuity of these partial derivatives):


 * $$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$$

and
 * $$\frac{\partial v}{\partial x} = - \frac{\partial u}{\partial y}$$

These equations are known as the Cauchy-Riemann equations.


 * Note: These equations are frequently written in the more compact form, $$u_x = v_y$$ and $$v_x = - u_y$$.

They may be obtained by noting that if the approach path is on x-axis then $$dz=dx$$, $$\partial f / \partial y$$ is irrelevant, so


 * $$\frac{df}{dz} = \frac{1}{2} \left ( \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} \right )$$

and that on the y-axis $$dz=i\,dy$$, $$\partial f / \partial x$$ is irrelevant, so


 * $$\frac{df}{dz} = \frac{1}{2} \left (-i \frac {\partial u}{\partial y} + \frac{\partial v}{\partial y} \right )$$

These equations have far-reaching implications. To get some idea of why this is so, consider that we can take second derivatives to obtain


 * $$u_{xx} + u_{yy} = 0$$

and
 * $$v_{xx} + v_{yy} = 0$$

In other words, u and v satisfy Laplace's equation in 2 dimensions. These functions arise in mathematical physics as scalar potentials in, for example, fluid dynamics. Laplace's equation is also basic to the study of partial differential equations. This is but one indication of the reason for the ubiquity of complex functions in physics.

Integration
By contrast, the definition of integration in complex analysis involves no surprises. Path integrals and integrals over regions are defined just as they are in the calculus of functions of two real variables. What is different is that the Cauchy-Riemann equations imply that integrals of complex functions have some very special properties. In particular, if a function f is differentiable (in the sense explained above) in a simply connected domain (intuitively, a domain having no "holes" in it), then for any closed curve $$\gamma$$ defined in that domain


 * $$\int_{\gamma}\nolimits f \, dz = 0.$$

It is essential that the domain of definition be simply connected. For example, let


 * $$D = \{ z \mid \textstyle\frac 1 2 < |z| < \frac 3 2 \}$$

and let $$f(z) = 1/z$$. Then if we define $$\gamma (t) = e^{it}$$ where t ranges from 0 to $$2 \pi$$ (i.e., we take $$\gamma$$ to be the unit circle), then the integral will not be 0.

It follows that if $$\gamma_1$$ and $$\gamma_2$$ are two homotopic paths joining a pair of points $$P, Q \in D$$ (intuitively, one can be continuously deformed into the other), then


 * $$\int_{\gamma_1}\nolimits f dz = \int_{\gamma_2}\nolimits f \, dz.$$

This is commonly expressed by saying that the integrals are path independent, and this is just the condition for the existence of a scalar potential!

Finally, we note that integrals in domains containing singularities (such as 1/z in the above example) can be computed using Cauchy's integral formula


 * $$f(z) = \frac{1}{2\pi i} \int_{\gamma}\nolimits \frac{f(\zeta) \, d \zeta}{\zeta - z}.$$

This result lies at the heart of many applications of complex analysis to disciplines ranging from number theory to physics. Its importance would be difficult to overestimate.