Intereting Posts

$H_1\triangleleft G_1$, $H_2\triangleleft G_2$, $H_1\cong H_2$ and $G_1/H_1\cong G_2/H_2 \nRightarrow G_1\cong G_2$
Infinite solutions of Pell's equation $x^{2} – dy^{2} = 1$
Is the Lie bracket of a Lie Algebra of a Matrix Lie Group always the commutator?
How to make a smart guess for this ODE
Property of limit inferior for continuous functions
Composition of two reflections (non-parallel lines) is a rotation
Does it make sense to define $ \aleph_{\infty}=\lim\limits_{n\to\infty}\aleph_n $? Is its cardinality “infinitely infinite”?
Area of intersection between 4 circles centered at the vertices of a square
Expected Value of the maximum of two exponentially distributed random variables
Why is “for all $x\in\varnothing$, $P(x)$” true, but “there exists $x\in\varnothing$ such that $P(x)$” false?
Is the integral of a measurable function measurable?
Integers as a sum of $\frac{1}{n}$
Induction Proof: Fibonacci Numbers Identity with Sum of Two Squares
If $a^m-1 \mid a^n -1$ then $m \mid n$
Why the $O(t^2)$ part in $L(t) = L + t(\csc \alpha_i – \cot \alpha_i +\csc \alpha_{i+1} – \cot \alpha_{i+1}) + O(t^2)$?

This picture

from Visual Complex Analysis is all you need to derive the Cauchy-Riemann equations, i.e. from the picture we see $i \frac{\partial f}{\partial x} = \frac{\partial f}{\partial y}$ should hold so we have

- Show that if f is analytic in $|z|\leq 1$, there must be some positive integer n such that $f(\frac{1}{n})\neq \frac{1}{n+1}$
- Difference between $\mathbb{R}$-linear and $\mathbb{C}$-linear maps
- Modification of Schwarz-Christoffel integral
- Prove that a complex-valued entire function is identically zero.
- Any linear fractional transformation transforming the real axis to itself can be written in terms of reals?
- Special biholomorphic mapping from $ \mathbb{C} \setminus \{z : z \le 0\}$ to the unit disk

$$i \frac{\partial f}{\partial x} = i \frac{\partial (u+iv)}{\partial x} = \frac{\partial (u+iv)}{\partial y} \rightarrow C \ R \ Eq’s$$

Is there a similar picture-derivation of the operators

$$\frac{\partial}{\partial z} = \frac{1}{2}(\frac{\partial }{\partial x} – i\frac{\partial }{\partial y})$$

$$\frac{\partial}{\partial \bar{z}} = \frac{1}{2}(\frac{\partial }{\partial x} + i\frac{\partial }{\partial y})?$$

The fact the differential forms can be visualized in terms of sheets tells me there can be one, any ideas?

- Does constant modulus on boundary of annulus imply constant function?
- Complex polynomial of degree $n$
- Laurent Series for $\cot(\pi z)$
- How does one define the complex distribution $1/z$?
- Proving that $\phi_a(z) = (z-a)/(1-\overline{a}z)$ maps $B(0,1)$ onto itself.
- All the zeroes of $p(z)$ lie inside the unit disk
- Bound on $|f'(0)|$ of a holomorphic funtion in the unit disk
- Why can't an analytic function be injective in a neighborhood of an essential singularity?
- Morera's theorem of entire function
- Show that $f^{n}(0)=0$ for infinitely many $n\ge 0$.

Here is a trigonometric explanation that is intuitive for me.

Your initial basis vectors are $2$ vectors of length $1$, $x=(1,0)$ (parallel to $OX$ axis) and $y=(0,1)$ (parallel to $OY$ axis). This is what you have on your left hand side zoom picture.

If you turn them by $45$ degrees clockwise (NOT as shown on your right hand side zoom picture, in the opposite direction), you will have $z=\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}(x+iy)$ and $\bar{z}=\left(\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}(x-iy)$ vectors respectively. The corresponding differentials will be

$\frac{d}{dz} = \frac{1}{\sqrt{2}}\left(\frac{d}{dx}-i\frac{d}{dy}\right)$ and

$\frac{d}{d\bar{z}} = \frac{1}{\sqrt{2}}\left(\frac{d}{dx}+i\frac{d}{dy}\right)$ respectively. For this rotation, we use http://en.wikipedia.org/wiki/Rotation_matrix, $\theta=\frac{\pi}{4}$.

If you want to use $x\pm iy$ as a basis, you are stretching the basis defined in the previous paragraph by $\sqrt{2}$ and so you have to stretch the differentials by $\frac{1}{\sqrt{2}}$ to compensate, $$\frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}}\left(\frac{d}{dx}\pm i\frac{d}{dy}\right) = \frac{1}{2}\left(\frac{d}{dx}\pm i\frac{d}{dy}\right) $$

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- What is the sum of this series? $\sum_{n=1}^\infty\frac{\zeta(1-n)(-1)^{n+1}}{2^{n-1}}$
- How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?
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- Ideal in a matrix ring
- Help me solve a combinatorial problem
- Is Complex Analysis equivalent Real Analysis with $f:\mathbb R^2 \to \mathbb R^2$?
- Proof of $\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}$
- Find the derivative of $\sqrt{x}$ using the formal definition of a derivative
- Is it known or new?
- Sum: $\sum_{n=1}^\infty\prod_{k=1}^n\frac{k}{k+a}=\frac{1}{a-1}$
- Good exercises to do/examples to illustrate Seifert – Van Kampen Theorem
- Finding a specific improper integral on a solution path to a 2 dimensional system of ODEs
- $\ln(x^2)$ vs $2\ln x$