About the gamma function , math , and language dictionaries

I have a good knowledge of  English (see my literary books page) , and I have a number of  reliable language dictionaries I consult from time to time. I also know about science and mathematics (see my science books page and the problem solvers page) , and I sometimes find mathematical expressions or physics formulas interspersed between scientific definitions in dictionaries. But occasionally I have noticed that math expressions or formulas published in these reference works contain errors and inaccuracies or seem to have been published hastily and without enough care. Whether they have been published using a math writing software or html related coding , the lack of accuracy or completeness is visible.

One example from physics is that of the Schrodinger equation (HΨ = EΨ)  inserted in a scientific definition in a renowned English language dictionary .The summation sigma symbol ( ∑) in the expanded Schrodinger equation is shown followed by the index of summation  and by the lower and upper bounds of summation as if they were multiplied by it . At other times parentheses are lacking  or a plus (+) sign , a letter or variable is omitted.

In one known language dictionary there was an error in the integral definition of the gamma function. I will not name any dictionaries  here , but in any case the error remained for a few years and  was later corrected in newer editions. The gamma function is related to the factorial by :gamma-def-1

and is defined by the improper integral

gamma-true-defIn the dictionary the definition was :

gamma-changedSo the variable of integration changed from dt to dx . In the online version of the dictionary ,the same integral expression is shown with the correct variable of integration , but it is missing a minus (-) sign and a letter variable.

Out of curiosity , I had the idea to find out what the value of the integral would be with a change of variable or if more than one letter or parameter were changed.

For the integral above with dx as integration variable , exp(-t) is constant , and the solution of the integral is :

integ-sol-01A plot of the solution gives (done with Mathematica):


Here is a 3D plot of the absolute value of the function  fct-solution in the complex plane (with the help of Mathematica) :

A generalized solution of the integral with dx and with arbitrary bounds of integration is the following:

integ-sol-02Another definition of the gamma function is :

gamma-true-def-logIf we change the integration variable we get:


One could go on making variations to the gamma function definitions and finding the corresponding solutions.

That was an excursion caused by a mistake in a math definition in a dictionary . Errors can be sometimes found even in science dictionaries and they are often corrected , but I think  if language dictionaries are able to write mathematical expressions and formulas more carefully and clearly , they would become more accurate and subsequently more appreciated.

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