which is the inverse function of , that is, . This solution is discussed next. Explicit solution An explicit expression for the diode current can be obtained in terms of the Lambert W-function (also called the Omega function). A guide to these manipulations follows. A new variable is introduced as . Following the substitutions : and : rearrangement of the diode law in terms of w becomes: , which using the Lambert -function becomes . With the approximations (valid for the most common values of the parameters) and , this solution becomes . Once the current is determined, the diode voltage can be found using either of the other equations. For large x, can be approximated by . For common physical parameters and resistances, will be on the order of 1040. Iterative solution The diode voltage can be found in terms of for any particular set of values by an iterative method using a calculator or computer. The diode law is rearranged by dividing by , and adding 1. The diode law becomes . By taking natural logarithms of both sides the exponential is removed, and the equation becomes . For any , this equation determines . However, also must satisfy the Kirchhoff's law equation, given above. This expression is substituted for to obtain , or . The voltage of the source is a known given value, but is on both sides of the equation, which forces an iterative solution: a starting value for is guessed and put into the right side of the equation. Carrying out the various operations on the right side, we come up with a new value for . This new value now is substituted on the right side, and so forth. If this iteration converges the values of become closer and closer together as the process continues, and we can stop iteration when the accuracy is sufficient. Once is found, can be found from the Kirchhoff's law equation. Sometimes an iterative procedure depends critically on the first guess. In this example, almost any first guess will do, say . Sometimes an iterative procedure does not converge at all: in this problem an iteration based on the exponential function does not converge, and that is why the equations were rearranged to use a logarithm. Finding a convergent iterative formulation is an art, and every problem is different. Graphical solution Graphical analysis is a simple way to