Integral expressions for 3d electric and magnetic potentials which are uniform in Euler terms and have non-integer orders of uniformity

In our previous paper “Universal expressions for 3D electric and magnetic potentials which are uniform in Euler terms” (see this issue) we already noted the usefulness of the paradigma of Euler’s uniform fields for designing of electron-optical and ion-optical systems. Non-integer orders of uniformity significantly extends possibilities and flexibility of usage for such fields in charged particle optics. As a result the analytical expressions for the Laplace potentials which are uniform in Euler terms are a useful tool to design optical systems of this type. However, at this moment general theory of harmonic and uniform 3D functions with non-integer order of uniformity is absent. This paper considers particular integral expressions which produces 3D harmonic functions which are uniform in Euler’ terms with non-integer orders of uniformity and can be used as potentials of corresponding electric and magnetic fields.