Wikipedia:Reference desk/Archives/Mathematics/2019 May 19

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May 19

Cauchy's integral formula proof

For a few days I am going over the proofs, but something just does not add up to me:

${\begin{aligned}&\oint \limits _{C}{\frac {f(z)}{z-z_{0}}}dz=\oint \limits _{C_{r}}{\frac {f(z_{0})}{z-z_{0}}}dz+\oint \limits _{C_{r}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}dz\\&\left|\oint \limits _{C_{r}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}dz\right|\leq \oint \limits _{C_{r}}{\frac {|f(z)-f(z_{0})|}{|z-z_{0}|}}dz=\oint \limits _{C_{r}}{\frac {|f(z)-f(z_{0})|}{r}}dz\\&\forall \varepsilon >0\exists \delta >0:\quad |z-z_{0}|<\delta \ \implies \ |f(z)-f(z_{0})|<{\frac {\varepsilon }{2\pi }}\\&\implies \ \left|\oint \limits _{C_{r}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}dz\right|<\varepsilon \end{aligned}}$

But no matter how I look at it, this is an equality of limits, not necessarily an equality of the integrals themselves! יהודה שמחה ולדמן (talk) 21:29, 19 May 2019 (UTC)

That's just for the case with no poles in the region, but if the limits are equal then the integrals must also be, or am I missing something? 67.164.113.165 (talk) 08:22, 20 May 2019 (UTC)

if the limits are equal then the integrals must also be, or am I missing something? You are not missing anything, but maybe יהודה שמחה ולדמן is.

The OP proves that

$\forall \epsilon >0,\left|\oint \limits _{C_{r}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}dz\right|<\epsilon$

The left-hand side of the inequality is a nonnegative number that is lower than any positive number (because crucially it does not depend on epsilon). The only such number is 0. If you are not convinced, assume it is equal to something positive, and take epsilon equal to half of it. Tigraan^{Click here to contact me} 09:09, 20 May 2019 (UTC)

Why does it not depended on $\varepsilon$? That makes no sense. For example, we agree that

$\left|{\frac {F(x+h)-F(x)}{h}}-f(x)\right|\leq \cdots <\varepsilon \ \implies \ \lim _{h\to 0}{\frac {F(x+h)-F(x)}{h}}=f(x)$

So why is this case different? יהודה שמחה ולדמן (talk) 10:57, 20 May 2019 (UTC)

It might help to know where you're getting this proof from; it's different from the one the WP article. As it stands the proof given is unclear on a number of points, but there aren't any limits in it (unless you count the integrals themselves as limits). First, I can't tell if the C in the first integral is meant to be C_r; if not then you need to apply the Cauchy's integral theorem to prove that the integral over C and the integral over C_r are the same, assuming that C_r is a circle with radius r and r is small enough that C_r is within C. Also, r needs to be smaller than δ for the estimate on |f(z) - f(z₀)| to be valid. It's actually not uncommon in analysis to use the argument outlined by Tigraan, but the argument has nothing to do with limits; it's really just a consequence of Trichotomy (mathematics). --RDBury (talk) 13:29, 20 May 2019 (UTC)

So why is this case different? Because you have left out the quantifiers. In this case, the quantifiers are $\forall \varepsilon >0$, whereas in the other case they are $\forall \varepsilon >0\exists \delta >0$. Quantifiers are very important! --JBL (talk) 15:48, 20 May 2019 (UTC)

Why don't we need $\delta >0$? יהודה שמחה ולדמן (talk) 16:55, 20 May 2019 (UTC)

Is this the way the proof goes?

${\begin{aligned}\oint \limits _{C}{\frac {f(z)}{z-z_{0}}}dz&=\oint \limits _{C_{r}}{\frac {f(z_{0})}{z-z_{0}}}dz+\oint \limits _{C_{r}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}dz\\\lim _{r\to 0}\oint \limits _{C}{\frac {f(z)}{z-z_{0}}}dz&=\lim _{r\to 0}\oint \limits _{C_{r}}{\frac {f(z_{0})}{z-z_{0}}}dz+\lim _{r\to 0}\oint \limits _{C_{r}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}dz\\\oint \limits _{C}{\frac {f(z)}{z-z_{0}}}dz&=\lim _{r\to 0}f(z_{0})\cdot 2\pi i=f(z_{0})\cdot 2\pi i\end{aligned}}$

1. The LHS integral is constant and independent of $r$.
2. The RHS integral sum is constant. (Duh?)
3. The first integral on the RHS is constant and independent of $r$.
4. The second integral on the RHS is constant, but we can also show it approaches 0 as $r\to 0$. Hence, it is 0.

Am I right? יהודה שמחה ולדמן (talk) 18:13, 21 May 2019 (UTC)

This version is certainly similar, but in step 4 the claim that the second integral →0 has no justification and that's really the whole crux of the argument. As I said above, the first version does not use limits, nor does it need to, and adding them in where they don't belong seems to be confusing things more. This is all assuming I've filled in the missing pieces of the first version correctly; I have no way of knowing since you still haven't given a source. It's very difficult to answer a question relating to what seems to be an incomplete and possibly garbled copy of something else. --RDBury (talk) 01:16, 22 May 2019 (UTC)

So why does this proof here use limits?

And again: Why do you keep saying we do not need limits, if we use $r<\delta$ which is the definition of a limit? יהודה שמחה ולדמן (talk) 10:20, 22 May 2019 (UTC)

Pretty sure that one proof using limits does not imply all proofs must use limits; different proofs of the same theorem can use different techniques. The proof at the start of this thread does use the definition of continuity, and you could say that continuity is defined in terms of limits (or that limits are defined in terms of continuity), but the proof uses the definition directly and does not use the $\lim$ or → symbols or any theorems about limits. Anyway, I'm not even sure what the original question was at this point. Is the original proof correct? Hard to tell since what you gave is apparently a copy with some details left out. Looking through my old textbooks I found one that appears to be the same: Erwin Kreyszig Advanced Engineering Mathematics 3rd Ed. pp. 534-535. This doesn't seem to be on-line anywhere, but [1] pp 41-43 is the same modulo wording and variable names. The proof is correct (though the last line of p43 of the pdf has questionable grammar), but if you claim it's incorrect because it uses limits somehow then you'll need to be more specific. --RDBury (talk) 19:12, 23 May 2019 (UTC)

As you said, I simply claim it is incorrect because

${\begin{aligned}\lim _{z\to z_{0}}f(z)=f(z_{0})\quad \equiv \quad \forall \varepsilon >0\exists \delta >0:\ |z-z_{0}|<\delta \ \implies \ |f(z)-f(z_{0})|<\varepsilon \end{aligned}}$

All this is only by assuming we are arbitrarily close to $z_{0}$. But if your circle is constant you cannot say anything about the integral other than it is constant.

This is why I say: The integral is constant, but we can also show it approaches 0 as $r\to 0$. Hence, it is 0. יהודה שמחה ולדמן (talk) 20:18, 23 May 2019 (UTC)

The statement

$\forall \varepsilon >0\exists \delta >0:\ |z-z_{0}|<\delta \ \implies \ |f(z)-f(z_{0})|<\varepsilon$

is true by the definition of continuity, there are no additional assumptions about z being close to z₀. If ε is given the you pick r to be smaller than the δ you get from continuity and also small enough that C_r is within C. Once you do this r is fixed, then it's a matter of applying inequalities to prove the integral has absolute value less than ε. So r depends on ε but it's not varying and there is no limit as r→0. This makes more sense if you've had a lot of practice with ε-δ proofs, and nothing I can write here can substitute for that practice, but I promise that when you've done it it should be clear that the proof is valid. (Actually, using limits without the ε's & δ's as a foundation is less correct. The symbol x→0 is supposed to mean the number x gets closer and closer to 0, but if you think about it a number is something like .213 and .213 is going to stay .213 no matter how hard you try to make it get closer to 0. Historically this kind of thing bothered a lot of people and it was ε's & δ's that resolved these issues.) --RDBury (talk) 10:31, 25 May 2019 (UTC)