I'm looking for the formula which converts WGS84 coordinates to EPSG:2163. I know that QGIS can do the calculations. but I need the exact formula?
EPSG: 2163 is a Lambert azimuthal equal-area projection, and you can easily find a brief idea of the required formulas on the wikipedia page for Lambert azimuthal equal-area projection. Please note that these are valid for a sphere.
Since we work with an ellipsoid, you need more intricate forumlas.
My go to reference for projection related information is Map Projections: A Working Manual, 1987, Snyder, John P.
It contains a lot of information about projections, includes detailed explanations, formulas, and examples. You should refer to it, for the detailed elipsoidal formula.
You can find the projection formulas in the above reference on pp 187-188 with an example on p. 332.