Question

General question about complex differentiation

Answer 1

Could you theoretically (even if practically this is stupid) think of complex differentiation in 'real terms' (By 'real terms' I mean treating z=x+iy, and i as a constant that squares to -1 when you simplify at the end)? It seems to work in that

f(z)=z^2
df/dz=2z,

Which is the same as

f(x,y)=(x+iy)^2
d(x+iy)^2/d(x+iy)=2(x+iy),

My question is basically: could you extend this to all complex differentiation, or was the above example a special case?

Answer 2

Yes, this is how it works. Basically, you are doing the same limiting procedure
as in the real case

df/dz = lim (f(z+h) - f(z))/h as h goes to zero.

What's a little trickier is that there are more directions for h to approach zero from,
as h is a complex number too.....

Answer 3

Thanks.
Is the gradient of any use with the multiple directions problem ('combining' them as I understand it), or is gradient just a useful thing in physics?

Answer 4

Yes and no. There's obviously an analogy between complex functions and
functions of two variables, but it isn't perfect. ( There are lots of non-physics applications of gradients, all you need is a problem two or more variables. for some reasom math people don't like to talk about it though....)