Directional derivative complex analysis
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I'm sorry if I sound too ignorant, I don't have a high level of knowledge in math.
The function f(z)=z2 (where z is a complex number) has a derivative equal to 2z.
I'm really confused about this. If we define the derivative of f(z) as the limit as h approaches 0(being h a complex number) of (f(z+h)−f(z))/h, then clearly the derivative is 2z, but what does this derivative represent??
Also, shouldn't we be able to represent a complex function in 4-dimensional space, since our input and output have 2 variables each (z=x+iy) and then we could take directional derivatives...right?
But if we define the derivative as above, it would be the same if we approach it from all directions. That's what's bothering me so much.
I would really appreciate any explanation. Thanks
The function f(z)=z2 (where z is a complex number) has a derivative equal to 2z.
I'm really confused about this. If we define the derivative of f(z) as the limit as h approaches 0(being h a complex number) of (f(z+h)−f(z))/h, then clearly the derivative is 2z, but what does this derivative represent??
Also, shouldn't we be able to represent a complex function in 4-dimensional space, since our input and output have 2 variables each (z=x+iy) and then we could take directional derivatives...right?
But if we define the derivative as above, it would be the same if we approach it from all directions. That's what's bothering me so much.
I would really appreciate any explanation. Thanks
Answered by
0
I'm sorry if I sound too ignorant, I don't have a high level of knowledge in math.
The function f(z)=z2 (where z is a complex number) has a derivative equal to 2z.
I'm really confused about this. If we define the derivative of f(z) as the limit as h approaches 0 (being h a complex number) of (f(z+h)−f(z))/h, then clearly the derivative is 2z, but what does this derivative represent??
Also, shouldn't we be able to represent a complex function in 4-dimensional space, since our input and output have 2 variables each (z=x+iy) and then we could take directional derivatives...right?
But if we define the derivative as above, it would be the same if we approach it from all directions. That's what's bothering me so much.
I would really appreciate any explanation. Thanks!
The function f(z)=z2 (where z is a complex number) has a derivative equal to 2z.
I'm really confused about this. If we define the derivative of f(z) as the limit as h approaches 0 (being h a complex number) of (f(z+h)−f(z))/h, then clearly the derivative is 2z, but what does this derivative represent??
Also, shouldn't we be able to represent a complex function in 4-dimensional space, since our input and output have 2 variables each (z=x+iy) and then we could take directional derivatives...right?
But if we define the derivative as above, it would be the same if we approach it from all directions. That's what's bothering me so much.
I would really appreciate any explanation. Thanks!
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