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$\int \frac{1}{x} dx$ 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?

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$d(1/x)/dx=-1/x^2$ ... – Jonathan Dec 23 '12 at 0:05
Oops, I'll fix that – hesson Dec 23 '12 at 0:06
I recognize that this is a losing battle, but: – Qiaochu Yuan Dec 23 '12 at 0:07
Anyway, the answer is "it depends on what you want to do." – Qiaochu Yuan Dec 23 '12 at 0:08
The logarithm function, and log of secant, predated the official discovery of the calculus. And the relationship between arc and chord was what ultimately gave birth to the sine. Opposite over hypotenus came much later. – André Nicolas Dec 23 '12 at 0:25

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.

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Another example along the same lines is the history of the Lambert-W Function. – Amzoti Dec 23 '12 at 0:32

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