Editing Differential Galois theory (section)

== Examples ==
* The field of rational functions of one complex variable '''C'''(''x'') becomes a differential field when taking the usual differentiation with respect to the variable ''x'' as the derivation. The field of constants of this field is the complex number field '''C'''.

* By [[Liouville's theorem (differential algebra)|Liouville's theorem]] mentioned above, if ''f''(''z'') and ''g''(''z'') are rational functions in ''z'', ''f''(''z'') is non-zero, and ''g''(''z'') is non-constant, then <math>\textstyle \int f(z)e^{g(z)} \, dz</math> is an elementary function if and only if there exists a rational function ''h''(''z'') such that <math>f(z) = h'(z) + h(z)g'(z)\,</math>. The fact that the [[error function]] and the [[sine integral]] (indefinite integral of the [[sinc function]]) cannot be expressed as elementary functions follows immediately from this property.
  
* In the case of the differential equation <math>y'' + y = 0</math>, the Galois group is the multiplicative group of complex numbers with absolute value 1, also known as the [[circle group]]. This is an example of a solvable group, and indeed, the solutions to this differential equation are elementary functions (trigonometric functions in this case).
  
* The differential Galois group of the [[Airy equation]], <math>y'' - xy = 0</math>, over the complex numbers is the [[special linear group]] of degree two, SL(2,C). This group is not solvable, indicating that its solutions cannot be expressed using elementary functions. Instead, the solutions are known as Airy functions.