It was then enough to consider the first order of a series expansion.

But, for the rest, we have to resort to series expansions!

This was the first power series expansion obtained in Wasan.

The above result can be derived by power series expansion of .

The singularities nearest 0, which is the center of the power series expansion, are at 2π"i".

The series expansion he obtained was effective at up to 85 degrees from the zenith.

This means that a meaningful Taylor series expansion in a neighbourhood of x is impossible.

We now use the formula above to give a Fourier series expansion of a very simple function.

The free energy curve is in excellent agreement with that obtained from high temperature series expansion.

It must be calculated either by numeric integration, or from a series expansion.

The following example, taken from page 11 of the system's manual, evaluates for using the Taylor series expansion.

For a similar proof with detailed series expansions, see .

Using that the series expansion of is given by:

For higher order poles, the calculations can become unmanageable, and series expansion is usually easier.

Taylor's series expansion of about is given below.

A single series expansion has been previously derived, but it makes use of nonexplicit functions.

This yields the infinite series expansion of the arctangent function.

Because squares to 1, the power series expansion of generates the trigonometric functions.

The photogeneration efficiency in this theory is usually expressed as a triple series expansion.

For many years, we have been using high-order perturbation series expansions to study these systems.

Again note that the series expansion provides a way to approximate the numerical value of the integral.

Be able to form power series expansions of simple functions and use them to construct approximations.

Still another way to evaluate this limit is to use a Taylor series expansion:

An alternative method is to use techniques from calculus to obtain a series expansion of the solution.

The Jackson integral of f is defined by the following series expansion: