# 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 $$ $$.