Power series solutions of polynomial equations passing through a specified point:

Plot the asymptotic solution and the solution it approximates:

Power series solutions of analytic equations passing through a specified point:

Plot the asymptotic solution and the solution it approximates:

Puiseux series solutions of polynomial equations passing through a specified point:

Solutions that are real valued when x approaches 0 from above:

Solutions that are real valued when x approaches 0 from below:

Puiseux series solutions of analytic equations passing through a specified point:

Asymptotic series solutions passing through a specified point:

Solutions of polynomial equations near a specified value of the independent variable:

Real asymptotic series solutions at infinity:

Plot the asymptotic solution and the solution it approximates:

Equations with symbolic parameters:

Conditions on parameters may be generated:

Power series solutions of polynomial systems passing through a specified point:

Plot the asymptotic solution and the solution it approximates:

Power series solutions of analytic systems passing through a specified point:

Plot the asymptotic solution and the solution it approximates:

Puiseux series solutions of polynomial systems passing through a specified point:

None of the solutions are real valued when t approaches 0 from above:

Two of the solutions are real valued when t approaches 0 from below:

Solutions of polynomial systems near a specified value of the independent variable:

Equations with symbolic parameters:

Power series solutions of polynomial equations passing through a specified point:

Plot the asymptotic solution and the solution it approximates:

Power series solutions of analytic equations passing through a specified point:

Plot the asymptotic solution and the solution it approximates:

Power series solutions of polynomial systems passing through a specified point:

Power series solutions of analytic systems passing through a specified point:

Power series solutions of polynomial systems near specified values of independent variables:

Specify conditions on parameters using Assumptions:

Different assumptions can produce different results:

By default, AsymptoticSolve gives solutions valid when x approaches 0 from above:

This finds the solutions valid when x approaches 0 from below:

Complex solutions may also depend on the direction:

This gives solutions that are real when x approaches 0 from a complex direction:

By default, AsymptoticSolve gives conditions it assumed to obtain the result:

This gives the result without the assumed conditions:

By default, assumed conditions that are generically true are not reported:

With GenerateConditions->True, all conditions are reported:

With GenerateConditions->None, AsymptoticSolve returns only generically valid results:

If nongeneric conditions are needed, AsymptoticSolve returns unevaluated:

Return a series whenever the result is a power series or a Puiseux series:

Check that the series solutions satisfy the equation:

By default, AsymptoticSolve[eqn,…,xa] computes the leading terms of the solutions:

Use SeriesTermGoal to obtain more terms:

The equation implicitly defines two different functions near each . Compute third-order asymptotic approximations for these two functions near :

Define functions based on these expansions:

At the point , these two functions have different values:

However, both exactly satisfy the equation at :

Visualize the equation and the approximations to its two branches:

In this case, it is easy to solve exactly for the two implicitly defined functions:

The two expressions returned by AsymptoticSolve are the series of the exact solutions:

Compute the second-order asymptotic approximations to the unit circle at :

The approximations closely track the circle of both larger and smaller values of at these regular points:

At the singular point , the approximation uses fractional powers:

Visually, the approximations seem to only be defined for values of :

This is because at the functions switch from being real to purely imaginary:

Trying to find solutions over the reals will therefore fail:

However, it is possible to find purely real expressions if restricting to smaller values of :

The curve crosses the line infinitely many times. On any section that passes the vertical line test—any vertical line intersects the curve only once; no vertical line intersects the section more than once—a function is implicitly defined:

Compute an approximation for the section that goes through the origin:

Note that this matches the Taylor series of , the inverse function of :

Compute an approximation for the section that goes through the point :

Visualize the curve and the two approximations:

Find solutions of a perturbed polynomial equation:

Plot the asymptotic solutions and the solutions they approximate:

Investigate the behavior of solutions of an analytic equation under a small perturbation:

Plot the asymptotic solutions and the solutions they approximate:

Find a series solution of an equation at a nonsingular point:

The derivative of with respect to does not vanish at :

Find the series solution up to order five:

The result satisfies the equation:

Find a multivariate series solution of a system of equations at a nonsingular point:

The Jacobian of with respect to does not vanish at :

Find the series solution up to order three:

The result satisfies the equations:

Find Puiseux series solutions in a neighborhood of a singular point of an algebraic plane curve:

Plot the asymptotic solutions and the curve they approximate near 0:

Find Puiseux series solutions in a neighborhood of a singular point of an algebraic space curve:

Plot the asymptotic solutions:

Find numeric solutions using asymptotic solution values as starting points:

Compare the numeric solutions and the asymptotic solutions:

Show the curve as an intersection of two surfaces:

Approximate Fermat's spiral near 0:

Compare plots:

Solve Kepler's equation for the eccentric anomaly in terms of the mean anomaly :

Compare with the exact solution for eccentricity :

Study the energy levels of a particle of mass in a one-dimensional box of width and depth . Solutions , and of the time-independent Schrödinger equation to the left of the box, inside the box, and to the right of the box are given by:

The solution must be continuously differentiable on the boundary of the box:

The homogenous linear equations admit nonzero solutions if their coefficient matrix is singular:

Assume that and are 1 and the units are chosen so that :

Find the possible energy levels for :

Compute the asymptotic solution near :

Compare the asymptotic solution and the minimum exact solution: