## 2010年8月14日星期六

### 網摘

1) 民進黨的粵語宣傳片

2) Can Intelligence Agencies Read Overwritten Data?

3) How to Read the Bible by James Kugel

4) 蕭愷一先生回應本博《蕭愷一錯評高考卷》一文

5) 講旅遊，不必講名勝

6) 爬蟲謎題

7) The Oppenheim's formula
This little-known cool trick can be attributed to a Prof. A. Oppenheim. Given the real part u(x,y) of an analytic function f(z) = u(x,y) + i v(x,y), it is a standard exercise in Complex Analysis to work out v(x,y) and f(z). The conventional solution is to apply the Cauchy-Riemann equations: since ux = vy and uy = -vx, we have
∫ ux dy = ∫ vy dy = v(x,y) + some function of x,
-∫ uy dx = ∫ vx dx = v(x,y) + some function of y.
Therefore, if we plug in the upper and lower limits of integration, we can determine v(x,y) up to an arbitrary constant.

In the above calculations, one needs to perform differentiation (on u) and integration (on the partial derivatives).  In the magical Oppenheim formula, however, you don't need to differentiate or integrate at all!  Here is the formula:
f(z) = 2 u( z/2, z/(2i) ) + c,
where c is a constant.

#### 2 則留言:

Thank you. It was most gracious of you to let my rebuttal be heard. I really appreciate that.

Hoiyat

don't have a strict prove for 6, just a framewoek:

1) any three dots travelling in non-parallel directions will tend to flatten as a straight line

2) For N>3 and non-convex, any non-convex angles will eventually flatten . So it became a convex paragon