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## Main Question or Discussion Point

I'm trying to understand the proof for this theorem:

Everything was going GREAT untill I got to this part:

I don't see this connection between "If [tex]U_{x}[/tex],[tex]U_{y}[/tex],[tex]V_{x}[/tex],[tex]V_{y}[/tex] are continuous at the point (x, y)" and

[tex]\Delta u = u_{x} \Delta x + u_{y} \Delta y + \epsilon_{1}| \Delta z|[/tex]

[tex]\Delta v = v_{x} \Delta x + v_{y} \Delta y + \epsilon_{2}| \Delta z|[/tex]

So everything after that is just moving deltas around... and I wish I knew why.

Where is this coming from? My book says it's a "famous result from analysis" but that just made me feel dumber for not knowing what it was. (I have not had real analysis, I'm taking it next term.) I looked in a real analysis book but I don't know what to look for... so that didn't work.

Can someone help me understand this step in the proof in a simple way that gets at the big idea behind the step? I have 16 more proofs to study for this midterm, so I don't want to get too bogged down... at the same time I don't want to resort to rote memorization that won't serve me well later when I take analysis and learn what the heck this is all about.

The function f(x) = u(x,y) + iv(x,y) is differentiable at a point z= x +iy of a region in the complex plane if and only if the partial derivatives [tex]U_{x}[/tex],[tex]U_{y}[/tex],[tex]V_{x}[/tex],[tex]V_{y}[/tex] are continuos and satisfy the Cauchy-Riemann conditions.

Everything was going GREAT untill I got to this part:

This shows that C-R is necessary, but now we much show that it is also sufficient: that is we must show that if the partials meet the C-R condition then f(z) is differentiable. Once we show this we will have proved the theorem.

If [tex]U_{x}[/tex],[tex]U_{y}[/tex],[tex]V_{x}[/tex],[tex]V_{y}[/tex] are continuous at the point (x, y) then:

[tex]\Delta u = u_{x} \Delta x + u_{y} \Delta y + \epsilon_{1}| \Delta z|[/tex]

[tex]\Delta v = v_{x} \Delta x + v_{y} \Delta y + \epsilon_{2}| \Delta z|[/tex]

Where [tex]| \Delta z|=\sqrt{\Delta x^{2}+\Delta y^{2}}[/tex]

[tex]\mathop{\lim}\limits_{\Delta z \to 0}\epsilon_{1} =\mathop{\lim}\limits_{\Delta z \to 0}\epsilon_{2}=0[/tex]

and

[tex]\Delta u = u(x+ \Delta x, y+ \Delta y)-u(x,y)[/tex]

[tex]\Delta v = v(x+ \Delta x, y+ \Delta y)-v(x,y)[/tex]

Calling [tex]\Delta f = \Delta u + i \Delta v[/tex], we have

[tex]\frac{\Delta f}{\Delta z}=\frac{\Delta u}{\Delta z}+i\frac{\Delta v}{\Delta z}[/tex]

[tex]=u_{x}\frac{\Delta x}{\Delta z}+u_{y}\frac{\Delta y}{\Delta z} + iv_{x}\frac{\Delta x}{\Delta z}+iv_{y}\frac{\Delta y}{\Delta z}+ (\epsilon_{1} +i\epsilon_{2})\frac{|\Delta z|}{\Delta z}[/tex]

... I was able to make sense of the proof from this point.

If [tex]U_{x}[/tex],[tex]U_{y}[/tex],[tex]V_{x}[/tex],[tex]V_{y}[/tex] are continuous at the point (x, y) then:

[tex]\Delta u = u_{x} \Delta x + u_{y} \Delta y + \epsilon_{1}| \Delta z|[/tex]

[tex]\Delta v = v_{x} \Delta x + v_{y} \Delta y + \epsilon_{2}| \Delta z|[/tex]

Where [tex]| \Delta z|=\sqrt{\Delta x^{2}+\Delta y^{2}}[/tex]

[tex]\mathop{\lim}\limits_{\Delta z \to 0}\epsilon_{1} =\mathop{\lim}\limits_{\Delta z \to 0}\epsilon_{2}=0[/tex]

and

[tex]\Delta u = u(x+ \Delta x, y+ \Delta y)-u(x,y)[/tex]

[tex]\Delta v = v(x+ \Delta x, y+ \Delta y)-v(x,y)[/tex]

Calling [tex]\Delta f = \Delta u + i \Delta v[/tex], we have

[tex]\frac{\Delta f}{\Delta z}=\frac{\Delta u}{\Delta z}+i\frac{\Delta v}{\Delta z}[/tex]

[tex]=u_{x}\frac{\Delta x}{\Delta z}+u_{y}\frac{\Delta y}{\Delta z} + iv_{x}\frac{\Delta x}{\Delta z}+iv_{y}\frac{\Delta y}{\Delta z}+ (\epsilon_{1} +i\epsilon_{2})\frac{|\Delta z|}{\Delta z}[/tex]

... I was able to make sense of the proof from this point.

I don't see this connection between "If [tex]U_{x}[/tex],[tex]U_{y}[/tex],[tex]V_{x}[/tex],[tex]V_{y}[/tex] are continuous at the point (x, y)" and

[tex]\Delta u = u_{x} \Delta x + u_{y} \Delta y + \epsilon_{1}| \Delta z|[/tex]

[tex]\Delta v = v_{x} \Delta x + v_{y} \Delta y + \epsilon_{2}| \Delta z|[/tex]

So everything after that is just moving deltas around... and I wish I knew why.

Where is this coming from? My book says it's a "famous result from analysis" but that just made me feel dumber for not knowing what it was. (I have not had real analysis, I'm taking it next term.) I looked in a real analysis book but I don't know what to look for... so that didn't work.

Can someone help me understand this step in the proof in a simple way that gets at the big idea behind the step? I have 16 more proofs to study for this midterm, so I don't want to get too bogged down... at the same time I don't want to resort to rote memorization that won't serve me well later when I take analysis and learn what the heck this is all about.