Suppose $\lambda$ and $c$ are real constants. I'm wondering if there's any special function that permits one to solve for the (real) variable $x$ in equations of the form $$x e^{-(x-\lambda)^2} = c$$ I'm familiar with the Lambert $W$ function and the solutions to the equation $x e^{-x^2} =c$, but I don't think that $W$ is involved in the general case I'm considering because of the presence $\lambda$, which alters the shape of the function $x e^{-(x-\lambda)^2}$ quite radically. Ideally, I'm looking for some sort of family of functions like $W$ that are parametrized by $\lambda$ -- does anyone know of references that might tackle this?
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If you are interested in solutions, then move the x to the other side: you then get a gaussian centered at $\lambda$ intersecting a hyperbola. Why do you need to know the name of the function? How will that help you solve the equation? Is there missing context here? – NickD Apr 06 '17 at 03:17
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@Nick I presume OP is looking for a Lambert $W$ style function particularly. – Cameron Williams Apr 06 '17 at 03:18
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@Nick The name is irrelevant -- I'm wondering if someone has already studied the properties of the function I seek; e.g., asymptotics, or if the solutions to the above equation can be written in terms of some other well-known special function. – sourisse Apr 06 '17 at 12:50