path integral methods. A summary of the development of a computer algorithm from the multislice theory of Cowley and Moodie for numerical computation was reported by Goodman and Moodie. They also discussed in detail the relationship of the multislice to the other formulations. Specifically, using Zassenhaus's theorem, this paper gives the mathematical path from multislice to 1. Schroedingers equation (derived from the multislice), 2. Darwin's differential equations, widely used for diffraction contrast TEM image simulations - the Howie-Whelan equations derived from the multislice. 3. Sturkey's scattering matrix method. 4. the free-space propagation case, 5. The phase grating approximation, 6. A new "thick-phase grating" approximation, which has never been used, 7. Moodie's polynomial expression for multiple scattering, 8. The Feynman path-integral formulation, and 9. relationship of multislice to the Born series. The relationship between algorithms is summarized in Section 5.11 of Spence (2013), (see Figure 5.9). Theory The form of multislice algorithm presented here has been adapted from Peng, Dudarev and Whelan 2003. The multislice algorithm is an approach to solving the Schrödinger wave equation: In 1957, Cowley and Moodie showed that the Schrödinger equation can be solved analytically to evaluate the amplitudes of diffracted beams. Subsequently, the effects of dynamical diffraction can be calculated and the resulting simulated image will exhibit good similarities with the actual image taken from a microscope under dynamical conditions. Furthermore, the multislice algorithm does not make any assumption about the periodicity of the structure and can thus be used to simulate HREM images of aperiodic systems as well. The following section will include a mathematical formulation of the Multislice algorithm. The Schrödinger equation can also be represented in the form of incident and scattered wave as: where is the Green's function that represents the amplitude of the electron wave function at a point due to a source at point . Hence for an incident plane wave of the form the Schrödinger equation can be written as We then choose the coordinate axis in such a way that the incident beam hits the sample at (0,0,0) in the -direction, i.e., . Now we consider a wave-function with a modulation function for the amplitude. Equation () becomes then an equation for the modulation function, i.e., . Now we make substitutions with regards to the coordinate system we have adhered, i.e., where . Thus , where is the wavelength of the electrons with energy and is the interaction