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It is of interest to show the connection with the [[Taylor series|Taylor expansion]] of an arbitrary potential around ''x''₀ Then the truncated Taylor series shrinks to

...rk conjectures]] predict that the coefficient of the leading term of the [[Taylor series]] expansion of an Artin L-function around ''s=0'' provides information abou

...can be found in the literature; both will be given below. The first is a [[Taylor series]] in Cartesian coordinates, while the second is in terms of [[spherical ha The [[Taylor series|Taylor expansion]] of a function ''v''('''r'''-'''R''') around the origin '

...''V'' over the charge distribution. ''V'' is then given by a two-term [[Taylor series|Taylor expansion]],

...y, we find that the principal branch of the Lambert ''W'' function has the Taylor series expansion

The [[radius of convergence]] of the Taylor series is unity, and the coefficient <math>g_n</math> does not decay as <math>n</m The Taylor series of <math>z!</math> developed at <math>z=1/2</math>, converges for all <math

...he ''harmonic approximation''&mdash;on which the method is based&mdash;the Taylor series is ended after this quadratic term. The second term, containing first deriv

// of the sum of the first 50 terms of the [[Taylor series]] for [[tetration]] developed at 3i.

The same [[Taylor series|series]] may be used to define the ''complex'' exponential function ...substituting <math>z = i\theta</math> and comparing terms with the usual [[Taylor series|power series expansions]] of <math>\sin \theta</math> and <math>\cos \theta

...ns|the significance of radians]] below.) One can then use the theory of [[Taylor series]] to show that the following identities hold for all [[real number]]s ''x'' ...o the trigonometric functions are defined on the complex numbers using the Taylor series above.

...''m'' = 1, the Jacobi matrix appears in the second (linear) term of the [[Taylor series]] of ''f''. Here the Jacobi matrix is 1 &times; n (the [[gradient]] of ''f

as can be verified using the [[Taylor series]] expansion:

The [[Taylor series]] for the tetration can be written in the usual form: The truncated Taylor series gives the [[polynomial approximation]].

...substituting <math>z = i\theta</math> and comparing terms with the usual [[Taylor series|power series expansions]] of <math>\sin \theta</math> and <math>\cos \theta

The diabatic potential energy surfaces are smooth, so that low order [[Taylor series]] expansions of the surfaces may be applied and the expansions do not intro

...generalized to higher powers in '''E''' (in the general case one uses a [[Taylor series]]), the polarizabilities arising as factors of E², and E<sup>3<

as can be verified using the [[Taylor series]]:

...hings he wrote about variational calculus and about the rest term of the [[Taylor series]] (1806).

...fferentiable]] function in this region. Expanding <math>f(x)</math> in a [[Taylor series]] around <math>x = x_k</math> gives

as can be verified using the [[Taylor series]]: