Xjyuk

Is there a theorem in Real Analysis similar to Cauchy's theorem/Cauchy's Integral Formula from Complex analysis?

If not, then why? What is it about the complex space that makes Cauchy's theorem true?

I actually think the real way to think about it is that Cauchy's Integral formula is a weird complex form of the Mean Value Property for harmonic functions.

Harmonic functions are characterised by the Mean Value Property, i.e., if $Usubseteq mathbbR^n$ is open, $xin U$ and $f$ is harmonic on $U$, then for all $r>0$ such that $B(x,r)subseteq U,$ we have

$$
f(x)=frac1dS(partial B(x,r))int_partial B(x,r) f(y)textrmdS(y),
$$
where $dS$ denotes the surface measure on $partial B(x,r)$. In case $n=2,$ this will be a regular cruve integral. The density appearing in the Cauchy formula needs to account for the fact that you're trying to compute a complex curve integral as opposed to a real one.

Note that holomorphic functions are, in particular, harmonic, so the above formula holds for them as a statement about real curve integrals.

The (possible) reasons are:

(1) If $f$ is analytic(differentiable) in $Bbb C$ then so is its derivative $f^'$ which is not true in $Bbb R$ , example consider the function $f(x)=x^2sin (frac1x)$.
(2) If you integrate an analytic function over a closed domain in $Bbb C$ (Cauchy's Theorem) then its integral is $0$ which is not true in $Bbb R$ , example consider $x^2 $ over $[-1,1]$

NOTE: For point (2) you could use the fact that every analytic function over a simply connected domain has an antiderivative which is not true in $Bbb R$

The essential difference between real and complex analysis at this level is that a complex function that's differentiable is analytic - it is the limit of its Taylor series in a disk. There are nonconstant infinitely differentiable real functions of a real variable all of whose derivatives at a point are $0$, hence not a power series in any disk.

See The distinction between infinitely differentiable function and real analytic function

Several theorems are named after Augustin-Louis Cauchy. Cauchy's theorem may mean:

Cauchy's integral theorem in complex analysis, also Cauchy's integral formula
Cauchy's mean value theorem in real analysis, an extended form of the mean value theorem.
Cauchy's criterion for uniform convergence.

Which Cauchy's theorem are you referring too?