-1
votes
2answers
36 views

Separating a Complex Valued Function

Is there a formula (with mathematical reasoning) for separating a complex-valued function $f(z)=f(x+iy)$ into the form $ f(z)=u(x,y) + iv(x,y)$? Thank You, C.A
0
votes
2answers
76 views

Real and imaginary parts of a complex-valued function

How do you get a complex-valued function $ f(z) = f(x+iy) = \frac{z^{s-1}}{e^{-z}-1}, $ where $s$ is a constant complex number and $z$ is a complex variable, into the form: $ f(x+iy) = a(x,y) + ...
0
votes
1answer
69 views

Separable Function: Alternative Representation

How does one get the following function $$ f(u) = f(x+iy) = \frac{u^{z-1}}{e^{-u}-1}, $$ where $z$ is a constant complex number and u is a complex variable, into the form: $$ f(x+iy) = v(x,y) + ...
1
vote
3answers
86 views

What is the real and imaginary parts for the complex function $f(z)=z^z$

I know: $f(z)=z^z =|z|^ze^{iz\theta} $ and $=|z|^z(\cos(zθ) + i\sin(zθ))$ But how do I continue to get the results for $\Re(z^z)$ and $\Im(z^z)$? $$\text { }$$ Thanks.
0
votes
0answers
24 views

Surjective complex map.

Show the map $(a + bi) \mapsto (a-bi)$ is surjective. Attempt: By definition, for every $(a - bi)$ in the complex set, there exists an $(a + bi)$ in the complex set such that $f[(a + bi)] = a - bi$. ...
1
vote
1answer
71 views

Conformal map entire domain to a strip with specific branchcuts

I am looking for a conformal mapping function that maps the entire z-plane to an infinite strip. (e.g. T=f(z) & -b < Real(T) > b ) I hope to find a function that cuts open to original domain ...
1
vote
2answers
39 views

Find real domain of a function results in $x \geq i$

I have an equation of the form $$f(x) = \sqrt{x^3 + x}$$ for which one needs to define the maximal domain, and image and domain are part of $\mathbb{R}$ (real numbers). $$x^3 + x \geq 0 \implies ...
1
vote
1answer
33 views

A complex log question

I'm trying to find the solutions of $\log(z)=i\log(\bar{z})$ where $\bar{z}$ is the conjugate of $z$. I'm aware of the multivalued complex log, so $\log(z)=\log|z|+i\arg(z)$ but I don't see to be ...
-1
votes
1answer
23 views

Which of the two points which the function is differentiable

$f(x+iy) = 2x + x^2y + i (7x^2/2 + 7y^2/2 -5y)$ Determine the two points where this function is differentiable. What is the real and imaginary part?
2
votes
0answers
43 views

Find a holomorphic function that satisfies this equation

Let $f:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ denote a function that satisfies the following equation: $$f(w,z+1)=w^{f(w,z)}$$ Is there a unique holomorphic function $f$ that satisfies this ...
0
votes
2answers
53 views

Are complex conjugates unique?

I'm trying to decide if a function $\varphi:\mathbb{C}\rightarrow \mathbb{C}$ defined by $\varphi(\alpha)=\bar{\alpha}$ is onto or one to one. I know it will be onto because every element in ...
4
votes
2answers
202 views

How to evaluate a zero of the Riemann zeta function?

Here is a super naive question from a physicist: Given the zeros of the Riemann zeta function, $\zeta(s) = \sum_{n=1}^\infty n^{-s}$, how do I actually evaluate them? On this web page I ...
2
votes
1answer
99 views

Complex logarithm and injectivity

Please forgive the trivial nature of this question: let U be a connected domain inside the punctured unit disk so that every curve inside it has winding number zero around the origin. Is the complex ...
0
votes
1answer
99 views

Linear independence of $\{\exp(b_{n}z):n\in\mathbb N\}$

I have the following question: Prove that the set $\{\exp(b_{n}z):n\in\mathbb N\}$ is linearly independent for some complex point $z\in\mathbb Z$. Prove that the set $\{\exp(b_{n}z):n\in\mathbb N\}$ ...
2
votes
1answer
227 views

Is there a geometric projection for every complex function

I was wondering about the best way to visualize complex functions. As they're $$ R^2 \rightarrow R^2$$ i think best way are complex plane image/grid transforms like they used in the Dimensions movie ...
2
votes
1answer
293 views

I want to to know how to show that this function is bijective between these sets [duplicate]

Possible Duplicate: Is this function injective and surjective? Let $f(x)=x^2$. In each of the following cases, is this function injective and/or surjective? $f: \mathbb{R} ...
2
votes
1answer
200 views

Is this function injective and surjective?

Let $f(x)=x^2$. Is this function injective and surjective if the function is defined as: $f: \mathbb{R} \longrightarrow [0,\infty)$. $f: \mathbb{C} \longrightarrow \mathbb{C}$. $f: \mathbb{R} ...
1
vote
2answers
571 views

Complex analytic functions and their zeros.

I am having trouble with the following statement found in a textbook: "Let $U$ be a connected open set. Let $f$ be a complex analytic function on $U$ and not constant. Either $f$ is locally ...
2
votes
2answers
231 views

A funny question, about the source of complex number.

As the video on http://www.youtube.com/watch?v=2kbM96Jr4nk It says that: $1\cdot(-1)=-1$ can represent as $1$ rotated $180^{\circ}$ around the Origin to get $-1$ $$1\cdot(-1)\mapsto ...
0
votes
1answer
57 views

Expressing ratio of functions as a constant number

Suppose that we have a function $f$ where $Q(\tau)$ is modified by multiplying $Q(\tau)$ by a real number. Let $f'$ be the modified function, and let $k \in \mathbb{R}$ or $k \in \mathbb{C}$. Take ...
4
votes
1answer
218 views

What is the Jacobian?

What is the Jacobian of the function $f(u+iv)={u+iv-a\over u+iv-b}$? I think the Jacobian should be something of the form $\left(\begin{matrix} {\partial f_1\over\partial u} & {\partial ...
2
votes
1answer
382 views

Image of a map in the complex plane

Is there an elegant way (either intuitive/ by a series of diagrams or by manipulating numbers/algebra) to find out what the image of $\sin(w)$ where $w\in \mathbb C$ from a domain say $\{w\in \mathbb ...
0
votes
1answer
341 views

If $\operatorname{Re}^{2}(x)=-1$, what is $x$?

$i=\sqrt{-1}$ $\operatorname{Re}(z)+i\cdot\operatorname{Im}(z)=z$ If $\operatorname{Re}^{2}(x)=-1$, what is $x$? $x$ cannot be defined in complex number as $(a+ib)$. { $a$ and $b$ are real numbers ...
0
votes
0answers
91 views

Uniqueness of homography

Let $(z_{1},z_{2},z_{3})$ and $(z'_{1},z'_{2},z'_{3})$ be two $3$-tuples of complex coordinates of non collinear points. How can I prove that there exists a unique homography $h$ such that ...
1
vote
2answers
318 views

Fixed points of $e^z$

How would one find the fixed points of $e^z$, where $z$ is complex (if there are any)? I feel this problem probably has a really obvious answer, and for some reason, I'm just not getting it. Thanks.
-3
votes
1answer
300 views

finding when a function is bijective/injective/surjective

Let $f:X\rightarrow X$ be defined by $f(x)=x^2$, where $X\subset \mathbb{C}$. What is an example of $X$ so that $f$ is bijective? Neither injective nor surjective? Surjective but not injective? ...
1
vote
2answers
424 views

How to express in closed form?

How to express this function in closed form without condition verfication and Re and Im functions (only with absolute value function)? $$f(z)= \begin{cases} - ...