# Glossary

• An ansatz is an expression for the general solutions of a problem, or for most of them, which typically involves undetermined coefficients, and is used for an educated guess.
• A Chebyshev expansion is a series of the form  where  denotes the th Chebyshev polynomial of the first kind.
• A formal power series  is an infinite sum of the form  where the  are in some common ring . The coefficient  of  in  is also denoted . The ring of formal power series is denoted .
• A formal logarithmic sum is a finite sum of the form  where the  are in some common ring  and the  are formal power series in .
• The indicial equation (or indicial polynomial) of a linear differential equation with polynomial coefficients is the coefficient of the power of_ with lowest exponent in the expression that is obtained by the substitution  into the equation. Its roots are exactly the exponents of  that can appear as lowest exponents in a series solution of the differential equation.
• A series  with nonnegative coefficients is a majorant series of a series  with complex coefficients if  for all .
• A complex number  is an ordinary point of a linear ordinary differential equation with polynomial coefficients, if the leading coefficient of the equation does not vanish at .
• The ramification index is the smallest positive integer  such that the exponent of an exponential factor can be expressed as a polynomial in .
• A complex number  is a singular point (or singularity of a linear ordinary differential equation with polynomial coefficients, if the leading coefficient of the equation vanishes at .
• The Laplace transform of a function  of the variable  is the function  defined by the integral .