In deriving the necessary conditions for a function to be differentiable at a point , why do we get ? Would not it be ?
Fundamentally, the answer is the "chain rule". Since z= z+ iy, and
Crucial point: 1/i= -i.
Another way of looking at the Cauchy-Riemann equations is this: With f(z)= f(x+ iy)= u(x,y)+ iv(x,y), the derivative, at , is the given by the limit just as for real numbers. And, just like limits for real numbers, in order that the limit exist, we must get the same result approaching along any path. In particular, if we approach along a line parallel to the real axis, h is real so we have
[tex]= \frac{\partial u}{\partial x}+ i\frac{\partial v}{\partial u}
Approaching instead along a line parallel to the imaginary axis, h is imaginary and we can use "ih", with i real, instead. Now we have
and the important difference is that "i" in the denominator. It will cancel "i" in the v part and remember that 1/i= -i so
In order that the limit exist, those two must be the same. Equating real and imaginary parts,
and