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I'm facing this function:

$$f(x)=\frac{1}{x}$$

What I know is that the above equation is one of the simplest forms of "rational functions", where the numerator is $1$ and the denominator is $x$.

Is its name "harmonic function" too? I have seen it in some papers with unclear explanations!

Comparing $f(x)=1/x$ and $f(x)=(x_o)\exp(-Cx)$ which is an exponential decay function, both give negative rate of change with some shared features, what the main difference between them? advantages and disadvantages?

First one some times diverges to $\infty$, correct?

Thanks

Zev Chonoles
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John
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    Advantages and disadvantages for what? – user7530 Feb 16 '13 at 22:02
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    I think you are quite confused about what you are asking. 1) No, this is not a harmonic function. 2) There is no "advantages and disadvantages". They are different functions, that is their main difference. 3) No, $1/x$ as $x \to \infty$ never diverges to $\infty$. It goes to $\infty$ as $x \to 0^+$. It goes to $0$ as $x\to\infty$. If you mean $\sum_n 1/n$, where $n$ runs over the naturals, then yes, it always diverges. – George V. Williams Feb 16 '13 at 22:05
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    I would call $f$ simply the reciprocal function. – Greg Martin Feb 16 '13 at 22:08
  • Thanks Greg, but if you google on it you will find some papers name it as a harmonic function and the sum of it is named harmonic series ... (2) I mean for advantages and disadvantages is rational function vs. exponential decay function as a model for negative rate of change, which one is better? ... (3) I mean summmation of 1/x for x=1:inf will diverge even the last term of 1/x seems to be zero but the summation will goes to infinite – John Feb 16 '13 at 23:51
  • I just call it the inverse/reciprocal function. – Thomas Feb 16 '13 at 23:52

3 Answers3

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You are right: The function defined by $$ f(x) = \frac{1}{x} $$ is a rational function. Some might call this the Harmonic function, but I don't think that this is a good idea because there is a (probably) more common notion of a Harmonic function.

The graph of the function is a Hyperbola.

Now, we do also have something called a Harmonic series. The Harmonic series is the series: $$ \sum_{n=1}^{\infty} \frac{1}{n} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots $$

In the end I don't believe that there is a commonly used name for the function $f(x) = \frac{1}{x}$.

You ask what the advantages of the function $f(x) = \frac{1}{x}$ compared to the other function that you list is. It isn't clear what exactly you mean by this. A function is just a function. Only what you know what you want to do can you talk about one function having advantages.

If you for example wanted to model something that decreases, say you want to model the decrease in a population of sorts, then it might turn out that the exponential function "works best" as a model.

Thomas
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  • Thanks Thomas for your great answer .. I have answered to Greg about the comparison between f(x)=1/x & g(x)=(Xo)exp(-CX) .. and also you have already answered to me when you suggest the exponential function as the best model .. I have tested both of them on the MATLAB & the rational function show instability for x goes to huge number while the exponential function remains in its good distributed shape .. Thanks again – John Feb 16 '13 at 23:58
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This could fairly be called the reciprocal function (the function that sends every real number to its reciprocal), or the multiplicative inverse function. The exponential function $e^{-x}$ has more applications as a physical model for various reasons, including that it remains bounded as $x\rightarrow0$.

Doubt
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The name is $f{}{}{}{}{}{}{}{}{}{}{}{}{}$

Will Jagy
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