AS engineer I would like that other applications to be mentioned (mainly linked to general laplacien type : electro mag potentials, wave equation, underground hydraulics,.. etc.,) and mainly that the research of a solution seemed to be triggered from the helmholtz eqn. Also please explain more about practically what is near 0. Sorry, I overlooked too hastily but I didn´t grasp the "ln" solution case (maybe too tired today). Please also, if legally allowable, mention the source of the series expansion demonstrations. Otherwise structured and clear but short.