Finding the complex structure which induces an almost complex structure (example) Given an almost complex structure (i.e. an endomorphism $\widetilde{A} : TM \rightarrow TM$ such that $\widetilde{A}^{2}=-$Id) defined by: $\widetilde{A}\left(\frac{\partial}{\partial x_{1}}\right) = e^{x_{2}}\frac{\partial}{\partial x_{2}}$ and $ \widetilde{A}\left(\frac{\partial}{\partial x_{2}}\right) = -e^{-x_{2}}\frac{\partial}{\partial x_{1}}$, what is the corresponding complex structure $A : M \rightarrow M$ (if it exists) which is induced by $\widetilde{A}$? 
Many thanks to all answers and comments, please ask if I need to give more information
 A: First, a couple of clarifications. You seem to be assuming that $M$ is a $2$-dimensional smooth manifold, and that it has a global coordinate chart with coordinate functions $(x_1,x_2)$; I'll assume this in what follows.
Next, I think you misunderstand the definition of a complex structure. Given a $2$-manifold $M$, a complex structure on $M$ is a covering of $M$ by coordinate charts $(U_\alpha,\phi_\alpha)$ taking their values in $\mathbb C$, such that where any two of them overlap, the transition map $\phi_\alpha\circ\phi_\beta^{-1}$ is holomorphic. A complex structure is induced by the given almost complex structure $\widetilde A$ if each coordinate map $z$ satisfies $Vz=0$ whenever $V$ is a $(-i)$-eigenvector of $\widetilde A$. 
Note that a complex structure is not a map from $M$ to itself.
With these understandings, you can approach the problem as follows. A little linear algebra shows that the $(-i)$-eigenspace of $\widetilde A$ is spanned at each point $(x_1,x_2)$ by the vector
$$
V = i\frac{\partial}{\partial x_1} - e^{x_2}\frac{\partial}{\partial x_2}.
$$
We're looking for a (locally defined) function $z$ satisfying $Vz=0$.  
The form of $V$ suggests trying for a coordinate function that's a real-valued function of $x_1$ alone plus an imaginary-valued function of ${x_2}$ alone: $z = u(x_1) + iv(x_2)$.
Then $Vz=0$ reduces to $i u'(x_1) - i e^{x_2}v'(x_2)=0$, which is equivalent to 
$u'(x_1) = e^{x_2}v'(x_2)$.  Since the left=hand side is a function of $x_1$ alone and the right-hand side is a function of $x_2$ alone, this equation can hold only if both sides are constants. Multiplying $z$ by a constant doesn't change the complex structure, so we might as well take the constant to be $1$. This leads to the solutions $u(x_1)=x_1$ and $v(x_2) = -e^{-x_2}$, or
$$
z = x_1 -ie^{-x_2}.
$$
By another stroke of luck, this coordinate function is globally defined and is a diffeomorphism onto its image.
So the desired complex structure is given by the single coordinate chart $(M,z)$, where $z(x_1,x_2)  = x_1 -ie^{-x_2}$.
