13
votes
0answers
98 views

How and why did Weierstrass $\wp$ get its special symbol?

I kind of always hated drawing the Weierstrass $\wp$ symbol by hand, and it struck me as odd how and why it achieved its special status in the first place. After all, there are tons of other important ...
1
vote
2answers
46 views

Uses of Jacobian of a map on $\mathbb{R}^n$.

For a map $f:\mathbb{R}^n\to\mathbb{R}^n$, Jacobian matrix of $f$ is defined as $$\begin{bmatrix} \frac{\partial f_1}{x_1}& \frac{\partial f_1}{\partial x_2}& \ldots \frac{\partial ...
2
votes
0answers
64 views

Who proved Fundamental Theorem of algebra using Liouville's theorem?

One of the most famous proofs of the Fundamental Theorem of Algebra involves Liouville's theorom stating that a bounded entire function in constant. Who first came up to the idea of deriving FToA ...
3
votes
1answer
268 views

sin(x) infinite product formula: how did Euler prove it?

I know that $\sin(x)$ can be expressed as an infinite product, and I've seen proofs of it (e.g. Infinite product of sine function). I found How was Euler able to create an infinite product for sinc by ...
1
vote
0answers
49 views

Weierstrass and Borel summation

In the Wikipedia article on Borel summation, there is the following quote attributed to Gösta Mittag-Leffler: Borel, then an unknown young man, discovered that his summation method gave the ...
0
votes
1answer
171 views

Square root principle value convention

Why is the principal square root of a complex number defined as $\sqrt z = \sqrt r e^{-i \varphi / 2}$ for $\varphi \in (-\pi, \pi]$ ? Wouldn't it be more natural to let $\varphi \in [0, 2\pi)$ as it ...
12
votes
1answer
162 views

Who is responsible for the analytical/topological proof of FTA?

The fundamental theorem of algebra asserts: Theorem Let $P$ be a polynomial of degree $\geq 1$ in $\Bbb C$. Then there exists a $z_1\in\Bbb C$ such that $P(z_1)=0$. The proof sketch goes as ...
1
vote
1answer
99 views

When was the significance of $i$ first noticed?

Complex analysis is an entire field of mathematics that focuses on the use of the complex constant $i$. When was the significance of $i$, an imaginary number, first noticed? If I did not know some ...
0
votes
1answer
83 views

“fluent” functions

In an old mathematics book (Ritt, 1948, p.5) I have come across the notion of "monogenic analytic" and "fluent" functions. These are complex valued functions. Has anyone heard of these terms before? ...
12
votes
2answers
225 views

Lie and Weierstrass' visualization of complex functions

I am reading Whittaker and Watson's A Course of Modern Analysis. In the third chapter where they discuss different ways to visualize functions that map the complex plane to the complex plane, they ...
5
votes
1answer
785 views

What applications of the Residue Theorem to real integration have had the biggest impact outside of pure math?

A typical undergraduate student (at least in North America) learns about integration of real-valued functions of one real variable, and learns some of its applications to science and probability, e.g. ...
4
votes
0answers
157 views

History Question - Branch Cut

My professor began discussing branch cuts in class today and mentioned that he did not know the origin of the term. Does anyone know the origin of the term and perhaps a source that talks about it?