**displaystyle**- By taking a = ∞ {\displaystyle \scriptstyle a\,=\,\infty } we normally recover the usual summation for convergent series. (wikipedia.org)
- The convergent version of summation for functions with appropriate growth condition is then: f ( 1 ) + f ( 2 ) + f ( 3 ) + ⋯ = − f ( 0 ) 2 + i ∫ 0 ∞ f ( i t ) − f ( − i t ) e 2 π t − 1 d t {\displaystyle f(1)+f(2)+f(3)+\cdots =-{\frac {f(0)}{2}}+i\int _{0}^{\infty }{\frac {f(it)-f(-it)}{e^{2\pi t}-1}}\,dt} To compare, see Abel-Plana formula. (wikipedia.org)
- The path of integration γ {\displaystyle \gamma } is counterclockwise around a closed, rectifiable path containing no self-intersections, enclosing c and lying in an annulus A in which f ( z ) {\displaystyle f(z)} is holomorphic (analytic). (wikipedia.org)
- Then, the posterior distribution of Θ {\displaystyle \Theta } is given by f ( Θ ∣ D ) = ∫ f ( Θ , G ∣ D ) d G , {\displaystyle f(\Theta \mid D)=\int f(\Theta ,G\mid D)dG,} where the integration represents summation over all possible gene tree topologies and integration over the coalescent times at each locus. (wikipedia.org)
- 2}{\tilde {\Phi }}(\mathbf {k} )} where k = m 1 b 1 + m 2 b 2 + m 3 b 3 {\displaystyle \mathbf {k} =m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}} in the final summation. (wikipedia.org)
- Once ρ ~ u c ( k ) {\displaystyle {\tilde {\rho }}_{uc}(\mathbf {k} )} is calculated, the summation/integration over k {\displaystyle \mathbf {k} } is straightforward and should converge quickly. (wikipedia.org)

**inhibitory**- In Hubel and Wiesel (1962), they reported that complex cells were intermixed with simple cells and when excitatory and inhibitory regions could be established, the summation and mutual antagonism properties didn't hold. (wikipedia.org)

**sums**- Ramanujan summation essentially is a property of the partial sums, rather than a property of the entire sum, as that doesn't exist. (wikipedia.org)
- In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. (wikipedia.org)

**Sequences**- Topics covered include differentiation techniques and applications, integration and some of its applications to physics and rate of change problems, sequences, series and matrices. (aber.ac.uk)

**infinite**- The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century, who thought of the integral as an infinite sum of rectangles of infinitesimal width. (wikipedia.org)
- Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. (wikipedia.org)
- Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined. (wikipedia.org)

**diffraction**- By rotating (precessing) a tilted incident electron beam around the central axis of the microscope, a PED pattern is formed by integration over a collection of diffraction conditions. (wikipedia.org)
- The result of this process is a diffraction pattern that consists of a summation or integration over the patterns generated during precession. (wikipedia.org)

**partial**- The continuous formulations include path integration, partial differential equations (in particular, the Schrödinger equation), and continuous optimization. (springer.com)

**method**- This formula originally appeared in one of Ramanujan's notebooks, without any notation to indicate that it exemplified a novel method of summation. (wikipedia.org)
- Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g., electrostatic interactions) in periodic systems. (wikipedia.org)
- The advantage of this method is the rapid convergence of the energy compared with that of a direct summation. (wikipedia.org)
- Ewald summation was developed as a method in theoretical physics, long before the advent of computers. (wikipedia.org)

**differentiation**- Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other. (wikipedia.org)
- Roughly speaking, the operation of integration is the reverse of differentiation. (wikipedia.org)
- Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. (wikipedia.org)
- The theorem demonstrates a connection between integration and differentiation. (wikipedia.org)
- Use and apply integration and differentiation with some notion of the relevance of these topics to physics. (aber.ac.uk)

**standard**- Let us evaluate this integral using a standard convergence result about integration by series. (wikipedia.org)

**Parts**- The summation by parts formula is sometimes called Abel's lemma or Abel transformation. (wikipedia.org)
- Summation by parts is frequently used to prove Abel's theorem. (wikipedia.org)

**path**- A trigger signal is generated to activate the passive safety system in the event of an impact or collision accident of the motor vehicle if the evaluated signal exceeds a certain threshold value corresponding to a critical velocity in the case of a single integration or a critical path distance in the event of a double integration. (google.com)
- end{aligned}}} Since the series converges uniformly on the support of the integration path, we are allowed to exchange integration and summation. (wikipedia.org)

**formula**- Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. (wikipedia.org)

**multiple**- After integration over multiple precessions, many more reflections in the zeroeth order Laue zone (ZOLZ) are present, and as stated previously, their relative intensities are much more kinematical. (wikipedia.org)
- this terminology carries over to the summation of multiple terms. (wikipedia.org)

**series**- Borel summation Cesàro summation Divergent series Ramanujan's sum Bruce C. Berndt, Ramanujan's Notebooks, Ramanujan's Theory of Divergent Series, Chapter 6, Springer-Verlag (ed.), (1939), pp. 133-149. (wikipedia.org)

**Although**- Their receptive field is therefore a summation and integration of the receptive fields of many input simple cells, although some input is directly received from the LGN. (wikipedia.org)

**order**- It is commutative, meaning that order does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see Summation). (wikipedia.org)

**certain**- A line integral is defined for functions of two or three variables, and the interval of integration [a, b] is replaced by a certain curve connecting two points on the plane or in the space. (wikipedia.org)

**level**- On one level, these are merely humble exercises in integration or summation. (springer.com)
- Such basic cells and ADCs based on them have a number of advantages: high speed and reliability, simplicity, small power consumption, high integration level for linear and matrix structures. (spiedigitallibrary.org)

**function**- Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. (wikipedia.org)