The coefficients of the Laurent series expansions of real analytic functions

Letf be a real analytic function of a real variable such that 0 is an isolated (possibly essential) singularity off. In the existing literature the coefficients of the Laurent series expansion off around 0 are obtained by applying Cauchy's integral formula to the analytic continuation off on the complex plane. Here by means of a conformal mapping we derive a formula which determines the Laurent coefficients off solely in terms of the values off and the derivatives off at a real point of analyticity off. Using a more complicated mapping, we similarly determine the coefficients of the Laurent expansion off around 0 where now 0 is a singularity off which is not necessarily isolated.     

New solid state NMR techniques in coal analysis

Nuclear magnetic resonance (NMR) spectroscopy has become a valuable technique for the determination of the aromaticity of fossil fuels. Now, by exploiting ¹³C relaxation, intimate details of the structure of coal may be determined. This article briefly describes these new applications.     

Proton-ionizable crown compounds. 4. New macrocyclic polyether ligands containing a triazole subcyclic unit

A series of macrocyclic polyether (crown) ligands containing the proton-ionizable s-triazole subcyclic unit were prepared by reacting the 1-THP blocked 3,5-bis(chloromethyl)-1H-1,2,4-triazole with various oligoethylene glycols. The starting bis(chloromethyl)triazole is a vessicant and must be used with caution. Triazolo-18-crown-6 ( 5 ) formed stable complexes with barium, strontium, copper and benzylammonium cations but not with potassium or lithium. The crystal structure of 5 showed the triazole proton to be on nitrogen 3 which is outside the macroring cavity.     

Thermal rearrangements of spiro[2.4]hepta-1,4,6-trienes

Thermolysis of spiro[2.4]hepta-1,4,6-triene (1a) at 50 degrees C yielded bicyclo[3.2.0]hepta-1,3,6-triene (5), which dimerized in two different fashions to form cyclobutanes. The 1,2-dimethyl and 1-propyl derivatives of 1a also rearranged at 50 degrees C, but at a faster rate, each yielding a pair of cyclobutane dimers. The structures of these symmetrical dimers were investigated by 1D and 2D NMR and NOE difference spectroscopy. Ab initio calculations indicated that the two strained olefins 1a and 5 had comparable energies about 50 kcal/mol lower than norborna-1(7),2,5-triene, which was thus excluded as a reaction intermediate.