MaplePrimes - answers and comments on Question, derivative of complex function

Reference

Markiyan Hirnyk — Fri, 06 Jan 2012 10:31:07 Z

Similar questions were asked and answered a lot. For example, look at http://www.mapleprimes.com/questions/103354-Differentiating-With-Respect-To-A-Function

> z := Complex(x, y);

Complex(x, y)
> Physics:-diff(f(Complex(x, y)), Complex(x, y));

D(f)(Complex(x, y))

PS. More exactly

> eval(Physics:-diff(z^2, z), z = Complex(x, y));

2 Complex(x, y)
and in the general case

hm ...

Axel Vogt — Sat, 07 Jan 2012 02:14:17 Z

z := Complex(x, y); evalc(%);

                               x + y I

Ok, but what should diff be in a general situation, where the
function is not analytic?

g:= t -> Re(t) + Im(t)^2*I;

Then

g(z);
                                    2
                               x + y  I

and

Physics:-diff(g(z), z);
                                  0
Physics:-diff(g(Complex(x, y)), Complex(x, y));
                                  0

Does that make sense?

By definition

Markiyan Hirnyk — Sat, 07 Jan 2012 09:37:28 Z

@karamand Your question was unclearly formulated: there is a diifference between complex functions and complex-valued functions. BTW, your previous question at http://www.mapleprimes.com/questions/96961-Plotting-A-Function-Versus-A-Geometric-Shape was unclearly formulated too. As far as I remember it from my student years (for example, see classic L. Hörmander, An introduction to complex analysis in several variables, Van Nonstrand Inc., Princeton (1966), Ch.1, 1.1), let u(x,y) be a complex-valued function, which is one time continuously differentiable on an open subset Ω of the complex plane, then its derivative with respect to z=x+I*y is defined by the relation diff(u(x,y),z):=1/2*(diff(u(x,y),x)+1/I*diff(u(x,y),y)). The one can be calculated with Maple. For example, consider U:=x^2 +y^3+I(x-sin(x*y). Then diff(U,z):= 1/2*(diff(U,x)+1/I*diff(U,y))=1/2*(2*x+I*(1-cos(x*y)*y)+1/I*(3*y^2-cos(x*y)*x))=x+I*(1-3*y^2-cos(x*y)*y+cos(x*y)*x)/2.

Regard, Markiyan Hirnyk

thanks

karamand — Fri, 06 Jan 2012 21:44:30 Z

which begs the question of how to best search Mapleprimes. I made several attampts before posting, using quotes and no quotes. I either got no hit or thousands of hits

thanks again

Search

Markiyan Hirnyk — Fri, 06 Jan 2012 22:28:06 Z

@karamand These links can be found by the "derivative with respect to function" search in MaplePrimes at the top of this page.

still negative

karamand — Sat, 07 Jan 2012 04:13:06 Z

@Markiyan Hirnyk

Sorry, still don't get it. If I search for the string in quotes I am getting zero(0) hits. If I leave it out I am getting abunch none germain to my question.

Now I have not used Physics before so based on the last post does it mean it works for analytic functions only.

And lastly suppose you went to the exercise and found the the anlytic function f(z)=Complex(u(x,y),v(x,y)).

How do you go about expressing it in terms of z and not x and y?