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Angular part of magnetic and electric vector spherical harmonics. Red and green arrows show the direction of the field. Generating scalar functions are also presented, only the first three orders are shown (dipoles, quadrupoles, octupoles).
The function e (−1/x 2) is not analytic at x = 0: the Taylor series is identically 0, although the function is not. If f ( x ) is given by a convergent power series in an open disk centred at b in the complex plane (or an interval in the real line), it is said to be analytic in this region.
That is to say, ¯ = for f ∈ H ℓ and g ∈ H k for k ≠ ℓ. Conversely, the spaces H ℓ are precisely the eigenspaces of Δ S n −1 . In particular, an application of the spectral theorem to the Riesz potential Δ S n − 1 − 1 {\displaystyle \Delta _{S^{n-1}}^{-1}} gives another proof that the spaces H ℓ are pairwise orthogonal and ...
In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite. A simulation-based alternative to this approximation is the application of Monte Carlo simulations.
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a ...
SymPy is simple to install and to inspect because it is written entirely in Python with few dependencies. [4] [5] [6] This ease of access combined with a simple and extensible code base in a well known language make SymPy a computer algebra system with a relatively low barrier to entry.
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
HKDF-Expand acts as a pseudorandom function keyed on PRK. This means that multiple outputs can be generated from a single IKM value by using different values for the "info" field. HKDF-Expand works by repeatedly calling HMAC using the PRK as the key and the "info" field as the message.