[python] fatal error: Python.h: No such file or directory

Solution for Cygwin

You need to install the package python2-devel or python3-devel, depending on the Python version you're using.

You can quickly install it using the 32-bit or 64-bit setup.exe (depending on your installation) from Cygwin.com.

Example (modify setup.exe's filename and Python's major version if you need):

$ setup.exe -q --packages=python3-devel

You can also check my other answer for a few more options to install Cygwin's packages from the command-line.