Friday 28 October 2005 at 2:49 pm | In Articles | 1 Comment

The Cauchy-Riemann equations are one of the first results one comes across in Complex Analysis. A poster on S.O.S. Mathematics Cyberboard has pointed that that proofs like that at Cauchy-Riemann equations tend to take it for granted that if f(x+iy)=u+iv is analytic then the partial derivatives of u and v exist. Thus the proof at Cauchy-Riemann equations says

\displaystyle f^{\prime}(z)=\lim_{h\rightarrow 0}{\left[\frac{u(x+h,y)-u(x,y)}{h}+i\frac{v(x+h,y)-v(x,y)}{h}\right]} and then deduces that \displaystyle f^{\prime}(z)=\frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x}

Looking at various textbooks this omission seems to the norm. Even Ahlfors Complex Analysis says: We remark that the existence of the … partial derivatives … is implied by the existence of f^{\prime}(z)

One excellent book A First Course in Complex Functions by G.J.O. Jameson does give a proper proof of this result. It defines differentiability for u:A \to \mathbb{R}^2 (where A is a subset of \mathbb{R}^2) at a point (a,b) in the interior of A if there exists real numbers \lambda,\mu such that, given \epsilon>0, there exists \delta>0 such that, for all real h,k with \sqrt{h^2+k^2}< \delta, |u(a+h,b+k)-u(a,b)-(\lambda h + \mu k)|\le\epsilon \sqrt{h^2+k^2}

Putting k=0 shows that \lambda = \dfrac{\partial u}{\partial x}; similarly \mu = \dfrac{\partial u}{\partial y}

If f^{\prime}(a+ib)=\lambda +i\mu then, given \epsilon>0, there exists \delta>0 such that for all real h,k with |h+ik|<\delta

|f((a+h)-i(b+k))-f(a+ib)-(\lambda + i\mu)(h+ik)|\le \epsilon|h+ik|

and taking real parts

|u(a+h,b+k)-u(a,b)-(\lambda h - \mu k)|\le\epsilon \sqrt{h^2+k^2} from which it follows that \dfrac{\partial u}{\partial x} and \dfrac{\partial u}{\partial y} exist. Taking imaginary parts gives the other 2 partial derivatives.


Saturday 22 October 2005 at 9:17 pm | In Articles | 6 Comments

Along with other British mathematicians I am very lax about how I write the multiplication symbol. I will write

    \1 \times 2 \
1.2 \
1 \cdot 2

and then maintain that the context will tell the reader what is meant.

The problem comes from the fact that:
1. British students aren’t used to using the dot as a multiplication symbol – it doesn’t seem to be used in schools. They only want to use \times

2. The international students have never used \times (though it appears on their calculators which seem to be designed for the American market) and they are much more careful with the dot which must be on the centre of the line. They also are unhappy about the British habit of not worrying where the decimal point goes: 1.2 or 1 \cdot 2

The students love to tell me off for using the ‘wrong’ symbol on the grounds that whatever I do is wrong 😎 It makes for very interesting discussions about the international differences in mathematics and led to a wonderful note from some of the students:

    If I insist on using . for multiplication then asking them to do Q.1 on p.77 should be read as Q \times 1 \text{ on } p \times 77 😕

It’s nice to leave a class with a smile on my face.

PS For many years British students have used the word ‘times’ as in times by 3 or even worse ‘timesing’. It sounds horrible to me but I seem to have lost the battle to say ‘multiply’

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