Is there any "real" use of complex analysis in quantum mechanics? After learning some quantum mechanics, I see a lot of applications of complex numbers. However, I have not yet seen any application of complex analysis. The full name for complex analysis is "functions of a complex variable", but in quantum mechanics, the wavefunctions are essentially functions from $\mathbb R\to\mathbb C$ (or $\mathbb R^3 \to \mathbb C$ if there are three dimensions), which does not have complex variables as their inputs.
The wikipedia article of complex analysis claims that there is some use of complex analysis in quantum mechanics. Is that actually true in the sense I have described above? To be more precise, can we see functions from $\mathbb C$ to $\mathbb C$ in quantum mechanics?
 A: One example: Complex analysis is used heavily in the proofs of the CPT theorem and spin-statistics theorem in relativistic quantum field theory. The classic book  


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*Streater and Wightman, PCT, Spin and Statistics, and All That
is filled with complex analysis, such as the "edge of the wedge" theorem described in Section 2-5.
This example isn't just some esoteric novelty; it is foundational to our current understanding of quantum physics. The CPT theorem is behind the existence of antiparticles, and the spin-statistics theorem is behind the Pauli exclusion principe, without which there would be no chemistry.
Complex analysis also comes up in conformal field theory, the study of quantum field theories with conformal symmetry. This is also fundamentally important, because scale-invariant quantum field theories (most of which are conformally invariant) are those to/from which all other quantum field theories flow when we zoom in/out. (This is an oversimplification, but the basic message is more important here than the caveats.)
Both of these examples involve quantum field theory, and one of the key properties of quantum field theory that makes complex analysis relevant is the combination of Lorentz symmetry with the spectrum condition, which says that the energy operator (Hamiltonian) must have a lower bound. This allows us to use Wick rotation, which is behind the use of complex analysis in both of the preceding examples. The spectrum condition is also fundamentally important: it ensures the existence of a vacuum state, which in turn is the basis for the definition of "particle" in quantum field theory.
So yes, complex analysis is important in quantum mechanics, even if this isn't evident early in the curriculum.
A: Yes, complex analysis does come up in many (relatively advanced) applications of quantum mechanics, most often in calculating the Green's functions for various differential operators. For example, Griffiths's undergraduate textbook uses it in section 11.4 to derive the Green's function for the Helmholtz equation, which is useful when studying the Born approximation. It's also used all the time in quantum field theory.
But for elementary quantum mechanics, you're correct that it doesn't come up very often because one typically considers complex wavefunctions whose domains are configuration space, not the complex plane.
