/sci/ - Science & Math

It seems like the magic that makes complex analysis tick boils down to the Cauchy Riemann equations. Here's a way to phrase them: Let [math]u,v: \mathbb{R}^2 \to \mathbb{R}[/math] and let [math]f = u + iv[/math]. Then [math]f[/math] is holomorphic iff it's real differentiable, and its derivative, when evaluated at any point, always takes the form
[math]\begin{pmatrix} a & -b \\ b & a \end{pmatrix}[/math]
with [math]a,b \in \mathbb{R}[/math]. Such matrices correspond to multiplication by a complex scalar, i.e. a composition of dilation and rotation.

QUESTION: What if we forget about the structure of [math]\mathbb{C}[/math] and generalize this to [math]n > 1[/math] dimensions? I.e. suppose [math]f: \mathbb{R}^n \to \mathbb{R}^n[/math] is (real) differentiable and assume also that its total derivative takes the above form when evaluated at any point (i.e. [math]\lambda T[/math] where [math]\lambda \in \mathbb{R}[/math] and [math]T[/math] is a rotation in [math]\mathbb{R}^n[/math]). Is there reason to believe that [math]f[/math] is (real) analytic?

What Kind of business can you start with a Maths degree? Also, how can you become successful with it by means of finding a way to make money with it?

>>8102118
one way to visualize complex functions is via colormap plots. the brightness indicates magnitude and the color indicates phase

>>7900014
I'm just having a hard time trying to interpret figures in the subject.

Look at this thing from wikipedia, for an example- It's supposed to be a complex function, with the hue representing the argument and the brightness representing the magnitude.

I just find it impossible to see any properties of the function using this, compared with a normal graph of a real function (where you could easily say "here it's differentiable" or "here is a singularity").

What is the most hand holding book on complex analysis?

Complex Variables and Applications (Brown and Churchill) is in my syllabus but it's too condensed and I'm just an engy.

Is calculus I-IV (includes multivariable and ODEs) along with some miscellaneous higher maths enough pre-requisite knowledge for Complex Analysis? No other analysis classes, but I'm slowly making my way through Rudin as well.