So we can replace the 7 with an ellipsis … to show that the process can be repeated indefinitely, thus proving the correctness of the equation. But we have to put in a strong caveat. As was pointed out by Barbara with a counterexample , the kind of infinite substitution we can do in this and the next problem are not necessarily valid in every case. Specifically, we need to be sure that the infinite expression is convergent. This is usually a task for a professional mathematician. We have to apply rigorous tests of convergence that can be applied in specific cases, as in the infinite nested radicals here.
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For each type, we can predict behaviors with such things as partial sum formulas. Another form of arithmetic progression, in the realm of continued fractions , is the following: where symbol corresponds to the Mathematica function ContinuedFractionK.
The geometric version of continued fractions is known as the Rogers—Ramanujan function R. In the lost notebook, F q represents S q. R q is a continued fraction of the form: and similarly for S q. The presence of the prefactor makes various formulas nicer. More formal definitions are as follows: These functions are related by. We can also define R and S in a way that can be evaluated more quickly through q-Pochhammer symbols. Here are pictures of the behavior of the R function on the unit disk in the complex plane.
Values returned can be complex, so these pictures show the imaginary, real, argument, and absolute values Im, Re, Arg, and Abs of the function R q. The unit circle itself is the natural boundary of analyticity and has a dense set of singularities of the function R q. As one can see, the Roger—Ramanujan functions are beautiful, not just due to their mathematical properties, but also visually. The functions R and S are two of the few named functions devoted to continued fractions.
That line at the end is equivalent to. Many of these have been found since Ramanujan wrote them down. All of these are readily solved with Mathematica. We list the values together with the first known solvers, with solutions by Oleg Marichev being first realized by Mathematica. We do not record the value here, because it is not particularly elegant. Ramanujan used chalk and his mind to simplify most of his results—the long results he erased from his slate, but the elegant results he wrote down.
First, calculate a numerical value for the point of interest. Second, conjecture a closed algebraic form for this number. Third, express the algebraic number as nested radicals. Finally, check the conjectured form with many digits of accuracy. Then we check that the numerical value of the conjectured form is the same as the value of the function. The values agree to at least places.
Since both of these are algebraic numbers with elegant representations, this is a rather convincing check. And the method can easily be generalized to find many more, so far unknown, values for S q , and similarly for R q.
An actual proof can be accomplished using modular equations. This is the modular equation of order 5 for S: We use the previously known value for for S q5 and solve for S q to obtain a value for. Clearing denominators, we obtain the above form of the result.
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Computable Document Format Computation-powered interactive documents. This is the modular equation of order 5 ramanujan unsolved problems S: These functions ramanujan unsolved problems related by. What a lovely tribute to the amazing mathematical genius, Srinivasa Ramanujam! A tout seigneur tout honneur! Questions Tags Users Badges Unanswered.
Solution: ‘Puzzles Inspired by Ramanujan’
Tygomuro We use the previously known value for for S q 5 and solve for S q to obtain a value for. An unelegant solution to the described problem which works in principle: Truly Ramanujan is the Mozart of Mathematics. Or continue as a guest your comment will be held for moderation:. These functions are related by. Several of the problems are elementary and can be attacked with a background of only high school mathematics.