Search results for "Trigo"
showing 10 items of 253 documents
A Lagrangian method for deriving new indefinite integrals of special functions
2015
A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral is derived which involves an arbitrary function, and therefore yields an infinite number of indefinite integrals for any special function which obeys such a differential equation. Techniques are presented to obtain the more interesting integrals generated by such an approach, and many integrals, both previously known and completely new are derived using the method. Sample results are given for Bessel functions, Airy functions, Legendre functions and hype…
Indefinite integrals involving the Jacobi Zeta and Heuman Lambda functions
2017
ABSTRACTJacobian elliptic functions are used to obtain formulas for deriving indefinite integrals for the Jacobi Zeta function and Heuman's Lambda function. Only sample results are presented, mostly obtained from powers of the twelve Glaisher elliptic functions. However, this sample includes all such integrals in the literature, together with many new integrals. The method used is based on the differential equations obeyed by these functions when the independent variable is the argument u of elliptic function theory. The same method was used recently, in a companion paper, to derive similar integrals for the three canonical incomplete elliptic integrals.
Probing the Multiple Structures of Vaterite through Combined Computational and Experimental Raman Spectroscopy.
2014
First-principles Raman spectra have been computed for several new vaterite structural models that have been recently proposed, and compared with spectra recorded on a set of biogenic, geological and synthetic samples. This set includes new measurements collected on Herdamania momus spicules (Great Barrier Reef, Queensland, Australia), which are known to have purity and crystallinity that are higher than for other biogenic samples. Overall, due to the close structural connection between the various models, the computed Raman spectra are found to be broadly similar. However, the spectra obtained for the two most stable models (monoclinic C2 and trigonal P3221, corresponding to two different p…
Crystal and molecular structure and electronic properties of [Cu(phen)2(HL)] · (phen)0.5 · 7H2O (H3L = 1,3,5-triazine-2,4,6(1H,3H,5H)-trione), a nove…
1998
Abstract The crystal and molecular structure of [Cu(phen)2(HL) · (phen)0.5 · 7H2O (H2L = 1,3,5-triazine-2,4,6(1H,3H,5H)-trione) has been determined by X-ray diffraction methods. The structure is built up of neutral [Cu(phen)2(HL)] entities (which are associated to form dinuclear units through hydrogen bonds involving the cyanurate ions), uncoordinated phenanthroline and crystallisation water molecules. The copper atom is involved in a CuN5 chromophore, with an stereochemistry near to a compressed trigonal bipyramid (TBP) but with a significant degree of distortion towards the square pyramidal (SP) topology. The EPR spectrum is indicative of a mixed d z 2 d x 2 −y 2 ground state for the copp…
Hypervalent tin-organic compounds: Vibrational spectroscopy in the solid as a tool for structure determination
1997
Abstract A full set of IR- and Raman spectra down from 600 cm1 has been determined and assigned for 60 penta- or hexacoordinated tin-compounds in the groups RSn(CH2CH2CH2)3N (R = Cl, Br, I, Me), RR'Sn(MCH2CH2)2X (R,R′ = Cl, Br, I, Me, Ph, M = CH2 S; X = NMe, O, S), RPh2SnCH2CH2CH2NMe2 (R = Cl, Br, I) and RMe2Sn—CHD—COOMe (R = F, Cl, Br, I, Me). In these hypervalent compounds an approach of the donor atom X to the central atom tin outlines a ‘path’ of nucleophilic attack from a tetrahedron to a trigonal bipyramid. Along this ‘reaction path’ the bond lengths of the axial ligand increase continuously while the bond lengths of the equatorial ligands slightly decrease. The number of similar hype…
Organotin(IV) chloride complexes with phosphocholine and dimyristoyl-L-?-phosphatidylcholine
2000
Several complexes of R n SnCl 4-n (R = Me, Ph, n = 1-3; R = nBu, n = 2, 3) with phosphocholine and dimyristoyl-L-α-phosphatidylcholine (phospholipid) have been synthesized and characterized by means of Mossbauer spectroscopy and NMR. Triorganotin chlorides form complexes of (R 3 SnCl) 2 .L stoichiometry with a trigonal bipyramidal pentacoordinate tin environment, while the others form 1:1 complexes with an octahedral hexacoordinate tin environment, with the ligands coordinating through anionic phosphodiester moieties in all cases.
Tin(IV) complexes with O-ethyl(N-ethyl-N,N-dimethylammoniomethyl)phosphonate
1999
Abstract O,O-Diethyl-(N,N-dimethylaminomethyl)phosphonate undergoes ethyl migration leading to O-ethyl-(N-ethyl-N,N-dimethylammoniomethyl)phosphonate (L). Several new complexes of tin(IV) and organotin(IV) chlorides with the title ligand have been synthesized. The stoichiometry of the obtained complexes is as follows: (R3SnCl)2 · L (R=Me, Bu, Ph), R2SnCl2 · L (R=Me, Bu, Ph), RSnCl3 L (R=Me, Ph) and SnCl4 · L. All the complexes have been studied in solution by means of 1H-, 13C-, 31P- and 119Sn-NMR spectroscopy. Their solid state structures have been investigated by means of Mossbauer spectroscopy and the molecular structure of the complex (Ph3SnCl)2 · L has been determined by X-ray crystall…
A Combined Experimental and Theoretical Study on Bis(μ‐alkoxo)diiron(III) Complexes with Hydroxybenzylaminoethanol [O,N,O] Donor Ligands: Syntheses, …
2011
Three new neutral bis(μ-alkoxo)diiron(III) complexes were prepared from N(R),N-(2-methylene-4,6-di-tert-butylphenol)aminoethan-1-ol ligands (H2L1, R = Me and H2L2, R = H). In these complexes, the ligand wraps around the metal center exhibiting a tridentate facial coordination mode with alkoxo-bridging oxygen, amine nitrogen and phenoxo oxygen donor atoms. In the complex [Fe(acac)L1]2·MeCN (1) acetylacetonato coligand complete the distorted octahedral coordination spheres of the iron(III) ions, whereas in the five-coordinate iron(III) chloride complexes [FeClL1]2 (2) and [FeClL2]2 (3) the ligands induce a geometry that is intermediate between square pyramidal and trigonal bipyramidal. Magnet…
New binary antimonide Hf5Sb3
1999
Abstract Hf5Sb3 can be prepared by arc-melting of hafnium and previously synthesized HfSb2. It crystallizes like the high-temperature modification of Zr5Sb3 in the Y5Bi3 structure type, space group Pnma, Z=4, a=740.75(9) pm, b=871.8(1) pm, c=1073.6(1) pm, V=693.3(1) 106pm3. An antimonide analogous to the low-temperature form of Zr5Sb3 was not obtained during our investigations. In the structure of Hf5Sb3, the Hf atoms form a three-dimensional network with numerous short Hf–Hf contacts, including the Sb atoms in severely distorted two- and three-capped trigonal prismatic voids. Calculations of the electronic structure, performed with the extended Huckel approximation as well as the TB-LMTO-A…
Ferrocenyl-functionalized tetranuclear gold(I) and gold(I)-copper(I) complexes based on tridentate phosphanes
2013
Tetranuclear AuI–FeII dimetallic and AuI–CuI–FeII trimetallic complexes bearing ferrocenyl (Fc) groups have been assembled by using two triphosphane ligands, namely, (PPh2CH2)2PPh (dpmp) and (PPh2)3CH (tppm). The compositions and structural type of the clusters are dependent on the stereochemistry of the P donor ligands. The complexes [tppmAu3Cu(C2R)3]PF6 [R = Fc (1) and 4-C6H4-Fc (2)] adopt a trigonal pyramidal {Au3Cu} arrangement of the coordinating metal core, whereas for the compounds with the linear triphosphane [Au4(dpmp)2(C2R)2](PF6)2 [R = Fc (3) and 4-C6H4-Fc (4)], a planar rhomboidal {Au4} framework was found. Clusters 1–4 were characterized by NMR spectroscopy and ESI-MS measureme…