This is a literature question, about possible proofs of some very basic results in complex analysis.

Some key facts about holomorphic functions are proved via reduction to smooth functions, using $\overline{\partial}$-techniques. For example, at the very foundation, we have the following result.

Theorem. There is a unique sheaf $\mathcal O$ ("the sheaf of holomorphic functions") on the topological space $\mathbb C$ such that for any open disc $D=D(x,r)=\{z\in \mathbb C\mid |z-x|<r\}$, one has $$ \mathcal O(D) = \{\sum_{n\geq 0} a_n (T-x)^n\mid \forall r'<r, a_n (r')^n\to 0\}, $$ the algebra of convergent power series on $D$ (with obvious transition maps). Moreover, for any open subset $U\subset \mathbb C$, one has $$ H^i(U,\mathcal O)=0 $$ for $i>0$.

The first statement says equivalently that a function $$ f: D\to \mathbb C $$ that admits, locally on $D$, a convergent power series expansion, in fact admits a power series expansion that is convergent on all of $D$. The usual way to prove this statement is to use Cauchy integrals to estimate the size of the coefficients $a_n$ in terms of boundary integrals. Weierstraß already wanted a proof that avoids integration, arguing more directly in terms of formulas; but to my knowledge, the only alternatives to this argument amount more-or-less to replacing the Cauchy integral by approximating finite sums.

Here is an elementary reformulation of (the essential content of) the second statement: Given any two open subsets $U,V\subset \mathbb C$, any holomorphic function on $U\cap V$ can be written as the sum of (the restrictions of) a holomorphic function on $U$ and a holomorphic function on $V$.

The usual way to prove the second statement is to use the resolution $$ 0\to \mathcal O\to C^\infty\xrightarrow{\overline{\partial}} C^\infty\to 0, $$ bump functions in $C^\infty$ to see that $H^i(U,C^\infty)=0$ for $i>0$ and another Cauchy integral to show that $\overline{\partial}$ is surjective after evaluating on $U$.

Question. Are there proofs that avoid integrals and non-holomorphic functions?

[Context: In my joint course with Clausen, we give such proofs, but I'm unsure what classical approaches to this fundamental result are known. Our motivation for doing so is that in $p$-adic geometry, neither integrals nor non-holomorphic functions make sense, so if we want complex and $p$-adic geometry to behave similarly, we should find a proof over the complex numbers that doesn't use those concepts.]

This is a literature question, about possible proofs of some very basic results in complex analysis.

Some key facts about holomorphic functions are proved via reduction to smooth functions, using $\overline{\partial}$-techniques. For example, at the very foundation, we have the following result.

Theorem. There is a unique sheaf $\mathcal O$ ("the sheaf of holomorphic functions") on the topological space $\mathbb C$ such that for any open disc $D=D(x,r)=\{z\in \mathbb C\mid |z-x|<r\}$, one has $$ \mathcal O(D) = \{\sum_{n\geq 0} a_n (T-x)^n\mid \forall r'<r, a_n (r')^n\to 0\}, $$ the algebra of convergent power series on $D$ (with obvious transition maps). Moreover, for any open subset $U\subset \mathbb C$, one has $$ H^i(U,\mathcal O)=0 $$ for $i>0$.

The first statement says equivalently that a function $$ f: D\to \mathbb C $$ that admits, locally on $D$, a convergent power series expansion, in fact admits a power series expansion that is convergent on all of $D$. The usual way to prove this statement is to use Cauchy integrals to estimate the size of the coefficients $a_n$ in terms of boundary integrals. Weierstraß already wanted a proof that avoids integration, arguing more directly in terms of formulas; but to my knowledge, the only alternatives to this argument amount more-or-less to replacing the Cauchy integral by approximating finite sums.

Here is an elementary reformulation of (the essential content of) the second statement: Given any two open subsets $U,V\subset \mathbb C$, any holomorphic function on $U\cap V$ can be written as the sum of (the restrictions of) a holomorphic function on $U$ and a holomorphic function on $V$.

The usual way to prove the second statement is to use the resolution $$ 0\to \mathcal O\to C^\infty\xrightarrow{\overline{\partial}} C^\infty\to 0, $$ bump functions in $C^\infty$ to see that $H^i(U,C^\infty)=0$ for $i>0$ and another Cauchy integral (in this context, a Cousin integral) to show that $\overline{\partial}$ is surjective after evaluating on $U$.

Question. Are there proofs that avoid integrals and non-holomorphic functions?

[Context: In my joint course with Clausen, we give such proofs, but I'm unsure what classical approaches to this fundamental result are known. Our motivation for doing so is that in $p$-adic geometry, neither integrals nor non-holomorphic functions make sense, so if we want complex and $p$-adic geometry to behave similarly, we should find a proof over the complex numbers that doesn't use those concepts.]