where h approaches z from some arbitrary direction to 0.
The trick is that in order to define a derivative it must be the same no matter what route we take to get h to go to 0. For example, let h approach 0 along the real axis:
But now let h approach 0 along the imaginary axis:
In order to evaluate this, let and the expression becomes:
Since the two answers are different we know that the derivative of cannot be defined.
".... of the form ...." means a function depending explicitly on and .
Egs. , , .
means take the partial derivative of wrt .
Egs. If then .
If then .
If then .
And of course, all complex functions of the form can be expressed in the form , in which case the more common formulation of C-R would probably be more easily applied ..... The formulation of Cauchy-Riemann I posted is rarely taught, in my experience,
(And please be kind enough not to retort with "And that's because it's ." )
and can make life easier or even trivial on occassion. A case in point is the question that created this thread.
I never took a course in complex analysis, so I really have no idea. But I can see why you would say that. I would assume that the majority of a basic course in complex analysis consists of complex-integration (which happens to be the fundamental concept).The formulation of Cauchy-Riemann I posted is rarely taught, in my experience,
I guess I better not post my proof .
I have seen the result a couple of times over the years in complex analysis textbooks - given as a question for the student to prove.
And don't be shy in facing the truth, TPH - progress in mathematics has always been due to physicists
It depends on what period. In ancient Greece, through the Medieval Ages, there was no physics and it was done only for math reasons. Later (at what I call) the Modern Age of math, Fermat and Descrates will build the fondation that will inspire Newton, in the 1600's. In the late 1600's and 1700's Newton began to create ideas of mathematics which were based on physics. This lead to studying differencial equations of mechanics. But soon the Bernoulli's, Lagrange, Fourier, and Euler (along with other mathematicians) added to what math used in physics. They developed more advanced methods in calculus, differencial equations, even began (Fourier) solving partial differencial equations, .... We are now at the 1800's. Close to 150 years the majority of the mathematics that is taught to engineering and physics students was devoloped during this time. Yes during this time much of mathematics we use today come from physics.
Fermat did not just no create a foundation of the mathematics that was mentioned above. There was another road in mathematics that he devoloped, a road is is not as commonly known as the one above. It was number theory (in fact he is called the father of modern number theory). Number theory is far far removed from any physical laws. It was exactly how math was done in ancient Greece, just for the sake of it. Fermat's problems inspired other mathematicians so solve them, and in effect create newer mathematics. People such as Gauss, Euler, Lagrange devoloped newer mathematics which was entirely removed from physics. This is an example, of a large area of math for which physics was not responsible for.
We should not forget the beginning of the 1800's as well. It was during this time when Galois, Abel, and others, were developing basic ideas for group and field theory. This area of math was again done for pure algebraic reasons with no connection to physics at all. It still contines to this date. And the amazing thing about this subject of math is that it even inspired physics (according to topsquark, concepts of field theory are used in quantum mechanics). This is again an area of math not influenced by physics at all.
There are more things to say, but I am not going to go there. For example, a lot mathematics of the 20th century and beyond is so far removed from the real world that it is not inspired by anything else.
It seems to me that without physics we would have had still much mathematics because by what said above. However, a big portion of mathematics would not be in existence today. Namely, analysis. After Calculus was formed mathematians where bothered by its informalities which lead them to a theory. Otherwise we might never had the same analysis as we have today.