Introduction to Special Functions

Olver, F. W. J.

doi:10.1016/b978-0-12-525856-2.50006-1

Cited by 86 publications

(145 citation statements)

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“…to the results of digital simulation), x 1. Thus, by using the asymptotic approximation 44 that exp(x) s (x) ≈ x s−1 in Equation 24, we arrive to the following approximative closed-form expression defining λ:…”

Section: E3176mentioning

confidence: 99%

Electrochemical Hydrogen Evolution: H⁺or H₂O Reduction? A Rotating Disk Electrode Study

Grozovski

Vesztergom

Ланг

et al. 2017

J. Electrochem. Soc.

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We study the effect of H + and OH − diffusion on the hydrogen evolution reaction in unbuffered aqueous electrolyte solutions of mildly acidic pH values. We demonstrate that the cathodic polarization curves measured on a Ni rotating disk electrode in these solutions can be modeled by assuming two irreversible reactions, the reduction of H + and that of water molecules, both following Erdey-Grúz-Volmer-Butler kinetics. The reduction of H + yields a transport-limited and thus, rotation rate-dependent current at not very negative potentials. At more cathodic potentials the polarization curves are dominated by the reduction of water and no mass transfer limitation seems to apply for this reaction. Although prima facie the two processes may seem to proceed independently, by the means of finite-element digital simulations we show that a strong coupling (due to the recombination of H + and OH − to water molecules) exists between them. We also develop an analytical model that can well describe polarization curves at various values of pH and rotation rates. The key indication of both models is that hydroxide ions can have an infinite diffusion rate in the proximity of the electrode surface, a feature that can be explained by assuming a directed Grotthuss-like shuttling mechanism of transport. A common problem of base metal electroplating is that the deposition of electroactive metals (e.g., Zn, Co, Fe or Ni) is almost inevitably accompanied by hydrogen evolution. In aqueous solutions hydrogen evolution always occurs if current is made high enough to exceed the limiting current of the deposited metal.1 In acidic solutions hydrogen may be formed as a result of H + reduction,while in solutions of pH > 7, the primary source of hydrogen is the electroreduction of water itself:Hydrogen evolution presents several ramifications for electrochemical plating processes. From a strictly technological point of view, the evolution of hydrogen is a serious concern because it reduces the faradaic efficiency of metal deposition, thereby increasing the amount of time and energy utilized to deposit the desired amount of metal at a given total current density. Another problem of hydrogen codeposition is that it can affect the structural and mechanical properties of the metal and can cause embrittlement. 1 Furthermore, hydrogen can also affect the kinetics of layer growth: if hydrogen adsorbs more efficiently on certain crystal planes of the metal, it may block these planes from further deposition so that the metal would preferentially grow on other planes.2 This inhomogeneous growth presents difficulties for the deposition of compact metal layers at the best -that is, if the accumulated gas does not impair the entire plating process. Finally, hydrogen evolution also results in the removal of H + ions from the diffusion layer. Increasing the near-surface pH may alter the intended chemical and electrochemical reactions at the interface. For example, if the solution at the interface becomes sufficiently alkaline, it may cause the solubility of a...

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Section: E3176mentioning

confidence: 99%

Electrochemical Hydrogen Evolution: H⁺or H₂O Reduction? A Rotating Disk Electrode Study

Grozovski

Vesztergom

Ланг

et al. 2017

J. Electrochem. Soc.

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“…The differential equation (9) expressed in z becomes the Legendre differential equation whose the theory in the complex plane is well known [13]. The two fundamental solutions in the complex plane are denoted P n (z) and Q n (z).…”

Section: Radial Functions Of the Multipole Solutionsmentioning

confidence: 99%

Electrostatics in a simple wormhole revisited

Boisseau

Linet

2013

Gen Relativ Gravit

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The electrostatic potential generated by a point charge at rest in a simple static, spherically symmetric wormhole is given in the form of series of multipoles and in closed form. The general potential which is physically acceptable depends on a parameter due to the fact that the monopole solution is arbitrary. When the wormhole has Z 2 -symmetry, the potential is completly determined. The calculation of the electrostatic self-energy and of the self-force is performed in all cases considered.

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“…At q = 1 we return to the sextic model where the "redundant" last row fixes one of the couplings and where we are left with a diagonalization of an N by N matrix which defines the N−plet of the real QES energies in principle. In such a setting, an important piece of an additional encouragement results from the well known possibility of a definition of new QES Hamiltonians by a change of variables r → x and ψ(r) → x const φ(x) in the Schrödinger equation [13,14]. Even when considered just in the most elementary power-law form, this change is defined by the prescription [15] …”

Section: Changes Of Variables and An Extension Of Applicability Of Thmentioning

confidence: 99%

“…In Hilbert space, their set is complete: This may be explained via oscillation theorems and characterizes the harmonic oscillator as exceptional. In such a setting, one should recollect all the similar (i.e., Coulomb and Morse) exactly solvable potentials, but one need not mention them separately as long as they are formally equivalent to our harmonic oscillator example after a simple change of variables [13].…”

Section: Introductionmentioning

confidence: 99%

New exact solutions for polynomial oscillators in large dimensions

Znojil

Yanovich

Gerdt

2003

J. Phys. A: Math. Gen.

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A new type of exact solvability is reported. Schrödinger equation is considered in a very large spatial dimension D ≫ 1 and its central polynomial potential is allowed to depend on "many" (= 2q) coupling constants. In a search for its bound states possessing an exact and elementary wave functions ψ (proportional to a harmonicoscillator-like polynomial of a freely varying, i.e., not just small, degree N), the "solvability conditions" are known to form a complicated nonlinear set which requires a purely numerical treatment at a generic choice of D, q and N. Assuming that D is large we discovered and demonstrate that this problem may be completely factorized and acquires an amazingly simple exact solution at all N and up to q = 5 at least.

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Introduction to Special Functions

Cited by 86 publications

References 0 publications

Electrochemical Hydrogen Evolution: H⁺or H₂O Reduction? A Rotating Disk Electrode Study

Electrochemical Hydrogen Evolution: H⁺or H₂O Reduction? A Rotating Disk Electrode Study

Electrostatics in a simple wormhole revisited

New exact solutions for polynomial oscillators in large dimensions

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Introduction to Special Functions

Cited by 86 publications

References 0 publications

Electrochemical Hydrogen Evolution: H+or H2O Reduction? A Rotating Disk Electrode Study

Electrochemical Hydrogen Evolution: H+or H2O Reduction? A Rotating Disk Electrode Study

Electrostatics in a simple wormhole revisited

New exact solutions for polynomial oscillators in large dimensions

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Electrochemical Hydrogen Evolution: H⁺or H₂O Reduction? A Rotating Disk Electrode Study

Electrochemical Hydrogen Evolution: H⁺or H₂O Reduction? A Rotating Disk Electrode Study