 Analysis is the branch of mathematics that focuses on functions. So far you've encountered real functions of one variable and real functions of many variables. Real analysis studies functions of real variables with real outputs. In complex analysis, well, it's complicated. So in principle, you could have a function of real variables that outputs to a complex number. But really there's only one important function of that type. f of xy equals x plus iy. What about functions with complex inputs and real outputs? There are at least four important functions of this type, and we've already seen them. If we let z be a complex number of the form x plus iy, we define the real part of z, the imaginary part of z, the absolute value or the modulus of z, and the argument of z. Now the most useful are functions with complex inputs and complex outputs. It's convenient to write such functions our usual way, so we might express that function as a formula. But it's useful to consider them as a sum of real and imaginary parts. And importantly, u of xy and v of xy are real valued functions. So for example, we can write as a sum of real and imaginary components f of z equals z plus the real part of z. So let z be x plus iy then. And so we can group the real and imaginary parts separately. u of xy is 2x and v of xy is y. And we can do that for any function we want. So for example, f of z equals z squared plus 3z plus 1. Expanding the real and complex parts of our function gives us then collecting the real and complex components separately gives us and we might take something like log of z. So remember to find log of z, we need to rewrite z in exponential form. So hitting it with a log gives us so in terms of x and y, our function is and we'll show off a little bit by simplifying log of a square root and separating our function into its real and complex components gives us By rewriting complex functions as the sum of real functions of two variables we can use everything we know about multi-variable functions. So moving to the complex plane makes everything harder before making it significantly easier. We'll take a look at that next.