= Solution
The two opens are $U_1=\mathbb C\times\mathbb C^*$ and $U_2=\mathbb C^*\times\mathbb C$; their intersection is $(\mathbb C^*)^2$. There is just one degree-one term in the <Čech cochain complex>, and no degree-two term. The <Čech coboundary> sends $(g_1,g_2)$ to $g_2-g_1$, so
$$
\boxed{\check H^1(\mathfrak U,\mathcal O_X)
=\frac{\mathcal O((\mathbb C^*)^2)}{\mathcal O(\mathbb C\times\mathbb C^*)+\mathcal O(\mathbb C^*\times\mathbb C)}.}
$$
The denominator denotes the sum of the restricted function spaces.
To make the quotient explicit, expand a <holomorphic function> on the intersection in a normally convergent two-variable <Laurent series>,
$$
h=\sum_{m,n\in\mathbb Z}a_{mn}z_1^mz_2^n.
$$
The terms with $m\geq0$ extend to $U_1$. Of the remaining terms, those with $n\geq0$ extend to $U_2$. Both subseries converge normally on their stated domains, by the coefficient bounds from the <Cauchy integral formula>. The unique remaining representative is the doubly negative part
$$
h_{--}=\sum_{i,j\geq1}a_{-i,-j}z_1^{-i}z_2^{-j}.
$$
No nonzero series of this form lies in the denominator, since a function on $U_1$ has no negative $z_1$ exponents and a function on $U_2$ has no negative $z_2$ exponents.
Writing $w_i=z_i^{-1}$, these representatives are exactly $w_1w_2F(w_1,w_2)$ with $F$ an entire <holomorphic function> on $\mathbb C^2$. To see that $F$ is entire, integrate for the coefficients on arbitrarily small product circles: $|a_{-i,-j}|\leq M(r_1,r_2)r_1^ir_2^j$. Choosing $r_iR_i<1$ gives absolute convergence for $|w_i|\leq R_i$, for every finite pair $R_i$. Conversely every such entire $F$ supplies a normally convergent representative on the intersection. Therefore \b[the quotient consists of convergent doubly negative Laurent series], not merely finite Laurent polynomials:
$$
\boxed{\check H^1(\mathfrak U,\mathcal O_X)\cong w_1w_2\mathcal O(\mathbb C_w^2).}
$$
This gives the <holomorphic first cohomology of punctured complex two-space>; for instance $1/(z_1z_2)$ represents a nonzero class.
For the comparison with <sheaf cohomology>, $U_1,U_2$ and their intersection are <Stein manifolds>. They can be realized as closed <complex submanifolds> of affine complex spaces by adding equations $z_iw_i=1$ for their nonzero coordinates. <Cartan theorem B> makes their higher cohomology with coefficients in the <sheaf of holomorphic functions> vanish. Thus the cover is acyclic for $\mathcal O_X$, and the <acyclic cover theorem> gives
$$
\boxed{\check H^1(\mathfrak U,\mathcal O_X)\cong H^1(X,\mathcal O_X).}
$$
The individual cover members are Stein; their union has the nonzero cohomology just computed.
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