9
votes
2answers
217 views

History of the matrix representation of complex numbers

It is well-known to many that $\mathbb{C}$ can be represented by matrices of the form $\left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$. For example, see this question or this ...
0
votes
1answer
57 views

Matrix representation of complex numbers in exponential form

Do there exist matrices M and P for this equation? Or perhaps M and P dont need to be matrices? I saw this and this question after googling which made me wonder about whether the exponential form of ...
2
votes
0answers
31 views

Accurate computation of arcsec near branch points

The direct numerical implementations of the usual definitions of the complex $\mathrm{arcsec}(z)=\arccos(1/z)$ and similar for $\mathrm{arccsc}(z), \mathrm{arcsech}(z), $ etc are not accurate near ...
1
vote
1answer
99 views

Book suggestion- complex analysis -conformal mapping.

I am studying complex analysis. And I am using J. Bak and D.J. Newman's book.(springer) And now my studying topic is conformal map. In addition to this book, I want to learn other book names which ...
2
votes
1answer
113 views

History of complex numbers

I'm interested in the history of complex numbers - their origin and their subsequent development. I'd be very interested if anyone can provide references for finding out more about this topic.
1
vote
3answers
110 views

Complex book suggestions

I take complex analysis course. And my instructor use -Bak and Newman's complex analysis book, springer. This book explains too fast and superficially. Please give me book suggestions which are the ...
3
votes
5answers
365 views

Rigorous Textbook for Introduction to Complex Numbers/Analysis?

Does anybody know where I can find a rigorous textbook on developing complex numbers/analysis? I'm currently working through Needham's Visual Complex Analysis, which is interesting but non-rigorous. ...
5
votes
1answer
195 views

Good texts in Complex numbers?

I have asked some members on chat about good text to study complex numbers , they recommended for example , "Visual Complex Analysis" by Needham and "complex analysis" by Steins. But, I look for a ...
0
votes
2answers
62 views

Reference request for derivatives of complex functions

I have been searching for reference for derivatives of complex numbers. All I found so far were texts that were too convoluted for me to grasp. I was (and still am) searching for a reference that is ...
2
votes
5answers
296 views

Fourier Analysis

I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function? Can someone provide me simple information ...
7
votes
1answer
158 views

Is there a complex variant of Möbius' function?

When you're dealing with arithmetic functions, you might have come across the classical Möbius' function $$ \mu(n)=\begin{cases} (-1)^{\omega(n)}=(-1)^{\Omega(n)} &\mbox{if }\; \omega(n) = ...
3
votes
1answer
2k views

Relationship between complex number and vectors

What is the relation between complex numbers and vectors? I have read in some places "a complex number a 2-dimensional vector". One can easily observe that $i\cdot i=-1$ in complex multiplication ...
3
votes
2answers
283 views

Require brilliant resources to self teach.

I'm far from the level of mathematical knowledge every user on this website posseses, however I am very much determined to get there as my love for mathematics increases. These are the topics: ...
1
vote
1answer
88 views

Differential Geometry for C^n

Does anyone know a good resource to read up on differential geometry for 2 complex dimensions with an anti-symmetric metric tensor?
3
votes
1answer
79 views

Classifying continuous characters $\epsilon:\mathbf{C}^\times\to \mathbf{C}^\times$.

I recently saw the following claim: Let $\mathbf{C}$ denote the field of complex numbers together with its usual topology. If $\epsilon:\mathbf{C}^\times\to \mathbf{C}^\times$ is a continuous ...
4
votes
2answers
110 views

inequality with roots of unity

Do you know proofs or references for the following inequality: There exists a positive constant $C>0$ such that for any complex numbers $a_1,\ldots,a_n$ $$ |a_1|+\cdots+|a_n| \leq ...
1
vote
1answer
121 views

Do these zeros have real part equal to $0$?

I guess this is a known result but I could not find it on the Internet. Consider these equations formed from the reciprocals of the divisors of $n$ raised to a complex number $s=a+ib$ : ...
14
votes
4answers
590 views

Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$

I was reviewing some matrices and found this interesting if $r = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ then $rr=-I$, also $$\exp{(\theta r)} = \cos\theta I + \sin\theta r$$ No wonder, the ...