When to create transcendental function to solve "unsolvable problem"? $\int \frac{1}{x} dx$ is an unsolvable problem using standard laws of Calculus (power rule) without the use of the function $f(x) = \ln x$ which was handcrafted by mathematicians to solve such problems. If we go back even further, the function $f(x) = \sin x$ was also a transcendental function used to describe the changing relationship between the arc and chord of a circle - it was not until 1682 that Leibniz proved that $\sin x$ was indeed not expressible as an algebraic function. Today, we still have expressions that can't be evaluated precisely such as $\int x^x dx$ because it cannot really be expressed as a function using the standard toolkit of algebraic and transcendental functions that we currently have. This begs the question, when is it appropriate for mathematicians to come up with new transcendental functions as solutions to "unsolvable problems" including but certainly not limited to the integral expression presented above?
 A: There are slight timetable issues in your argument.  
For example John Napier's logarithms appeared in 1614, John Speidell compiled a table on the natural logarithm in 1619, and Nicholas Mercator first called this a natural logarithm in 1668, all before Isaac Newton and Gottfried Leibniz published work on integral calculus.    
Similarly trigonometric functions appear in Greek, Indian and Islamic mathematics, all before calculus.
In terms of your question, you are free to define a "new function" as the solution to a particular question, at least if such a question is well defined and you are clear what you are doing.  Whether anybody else takes up the definition may depend on how useful it is.  
A: It's less that we "created" them, we simply gave them a name, and that as well simply to make life easier on us.
I think at the core of every named function, there's a mathematician who needs to use that function intensely, and as e.g. the integral depiction is kinda lengthy and rather hard to handle notationally, they define a new function.
Whether or not the name sticks is then simply determined by the influence of their research, in which they named the function. If it's a powerful result, people are bound to read it, and then you've got a high chance that your given name will stick. If not, well, then probably it's gonna be a name only for your paper.
So in other words: 
Name the function whenever it's convenient for you (and/or the reader). The circle of mathematicians, and the future will then decide for you whether your given name is worthy of the function
