Questions tagged [continued-fractions]

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Replace time fractional derivative approximations algorithm to another algorithm, but why the values of the approximations have not been calculated? [closed]

I am trying to replace fractional derivative approximations algorithm in another algorithm, but the values of the approximations have not been calculated. Please help to replace approximate correctly ...
0
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0answers
69 views

Why my code give empty plots? AND When I increase the number of mx , mt my code is running for a long time

In my code, two graphs are not displayed. what is the problem?Also when I increase the number of mx and mt, the code enters the running for a long time. The regularization problem is described as $D^{...
2
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2answers
90 views

Representing a general continued fraction in a list for ContinuedFraction

The initial mathematical problem is to solve for $x$ in: $$x = \sqrt{19} + \frac{91}{\sqrt{19} + \frac{91}{\sqrt{19} + \frac{91}{\sqrt{19} + \frac{91}{\sqrt{19} + \frac{91}{x}}}}}$$ (Notice the $x$ in ...
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0answers
97 views

How to implement statistics for the length of continued fractions of a result on MATHEMATICA

After successfully generating the reduced fraction of two coprime in interval (0,1] with the following; ...
9
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1answer
348 views

Using Mathematica to find an alternative continued fraction for $\zeta(5)$

Given the Riemann zeta function $\zeta(n)$. I. $x=\zeta(3)$ Using Euler's continued fraction formula, we can form $\zeta(3)$'s cfrac as, $$Ax+B = \cfrac{1}{v_1 - \cfrac{1^6}{v_2 - \cfrac{2^6}{...
8
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3answers
456 views

Continued fraction expansion of a rational function

Maybe I'm missing something, but what would be a good way to calculate the continued fraction expansion of a rational function $Z(s) = N(s)/D(s)$? The built-in function ...
2
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1answer
65 views

Understanding FromContinuedFraction

In the detail section of the documentation page for FromContinuedFraction we read that: FromContinuedFraction[{a1,a2,…,{b1,b2,…}}] returns the exact number whose ...
3
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0answers
58 views

ContinuedFraction: different result with different representation of argument

Why if I write: In[7]:= ContinuedFraction[3.15] FromContinuedFraction[%] N[%] I get: ...
2
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1answer
98 views

Deriving formulas for continued fraction expansions

A version of a continued fraction expansion of a rational number $r\in \mathbb Q$ is defined as \begin{align} r =[a_0,a_1,a_2,\ldots,a_k]= a_0 - \frac{1}{a_1 - \frac{1}{a_2 - \dots - \tfrac{1}{a_k}}} \...
5
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1answer
1k views

Negative Continued Fraction of a Rational

The $n^{\text{th}}$ negative continued fraction convergent $x_n$ of a positive real $x$ is computed by the nested function \begin{align} x_n = k_1 - \frac{1}{k_2 - \frac{1}{k_3 - \dots - \tfrac{1}{k_n}...
3
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1answer
89 views

Simplify multiple layers fractions

I have a complicated fraction ...
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5answers
731 views

How can I write Mathematica code for this continued fraction with alternating terms?

$$\varphi+\cfrac{1}{\varphi^{-1}+\cfrac{1}{\varphi+\cfrac{1}{\varphi^{-1}+\cfrac{1}{\varphi+\cdots}}}}$$ I saw this continued fraction on Facebook. I need the Mathematica code for this using ...
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2answers
65 views

How can we generate this more general form of the continued fraction? [duplicate]

How can I write Mathematica code for this continued fraction with alternating terms? How can we generate this more general form of the above-continued fraction? $$x+\cfrac{a}{y+\cfrac{a^2}{x+\cfrac{...
1
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1answer
76 views

difference equation and continued fractions

I'm interested in solving the following difference equation: $x[k-1]+(k^2+k+a)/x[k]=b$, $k=1,2,\ldots$, where $a,b$ are fixed positive numbers; let's say $x[1]=c>0$. Mathematica's ...
1
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2answers
158 views

Computing continued fraction

I want to build this infinite continued fraction $$F_{n}(x)= \cfrac{1}{1-x\cfrac{(n+1)^2}{4(n+1)^2-1}F_{n+1}(x)} $$ which gives for $n=0$ $$F_{0}(x)=\cfrac{1}{1-\cfrac{(1/3)x}{1-\cfrac{(4/15)x}{1-\...
2
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3answers
423 views

Truncate an infinite continued fraction at order 2000

I want to solve an equation which contains an infinite continued fraction $F(n)$. Then I must (obviously) truncate this continued fraction at $n=2000$. The problem here is that Mathematica does not ...
5
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2answers
194 views

How to calculate this function

Please, I would like to calculate this function which contains an infinite continued fraction ...
2
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1answer
66 views

Can Mathematica evaluate a continued fraction using Gauss's $K$ operator?

I don't know a whole lot about Mathematica, and this is a fairly uncommon notation, so here goes: How do I tell Mathematica to evaluate this? $$1+\underset{i=1}{\overset{\infty}{K}}\frac{(-1)^{i-1}}{...
1
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1answer
191 views

Writing code to produce continued fractions

I have been trying to create a continuing fraction to help prove the theory about any rational number: a/b being able to be written as a continued fraction where the remainders found in the Euclidean ...
3
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3answers
76 views

Potential bug : missing terms in ContinuedFraction

If I use 0.23 instead of 23/100, the last continued fraction coefficient (7) is not given: ...
3
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1answer
50 views

Unexpected behavior of ContinuedFractionK with function defined by SetDelayed

I cannot explain the following results: First, I define a function f. Then ContinuedFractionK imvolving ...
3
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2answers
184 views

Problems with ContinuedFractionK

I am interested in calculating the following series $$ \sum_{k=0}^{+\infty}{ \frac{ x^k }{\, a (a+1) \cdots (a+k) \,} } $$ using this continued fraction (I expect) equivalent form: $$ \cfrac{1}{a + ...
2
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1answer
317 views

Gauss Continued Fraction for Hypergeometric Functions

I would like to calculate the Gauss Continued Fraction for this particular Hypergeometric function: \begin{equation} _{2}F_{1}\left( 1-\frac{1}{p}, \frac{1}{p}; 1+\frac{1}{p}; x^p \right) \end{...