Derrida's magic trick

Calculus Level 4

Find $\large \int\limits_0^\infty \frac{e^{-t}-e^{-Zt}}{t} \, dt$

in terms of $Z$ .

e^{-Z}

Z\ln Z

e^{-Z}/Z!

\ln Z

3 solutions

Mvs Saketh
Mar 1, 2015

$\begin{aligned} I &= \int _{ 0 }^{ \infty }{ \frac { { e }^{ t }-{ e }^{ -Zt } }{ t } } dt\\ \frac { \partial I }{ \partial Z } &= \int _{ 0 }^{ \infty }{ { e }^{ -Zt } } dt \\ &= -\frac {e^{ -Zt }}{Z} \Bigr|^\infty_0 \\ \\ &= \frac { 1 }{ Z } \end{aligned}$

The solution of this differential equation is given by $I = \log Z + \text{const.}$

Applying the boundary conditions, available in the definition of $I$ , we find that $I(Z=1) = 0$ , so that $\text{const.} = 0$ . Thus $I = \log Z$ .

Nice solution! I did it using Laplace transform.

Hasan Kassim - 6 years, 3 months ago

Oh, wow! My method was little different! I considered the taylor series rather. I didn't know about the "differentiating under the integral sign"

Kartik Sharma - 6 years, 3 months ago

It is a common trick, and sometimes very useful as here

Mvs Saketh - 6 years, 3 months ago

Can anyone comment on the name of this problem "Derrida's magic trick" as I'm not sure how to contact @Josh Silverman (although using the @ just now may have worked--sorry, I'm new here). Basically, I'm a grad student in English who studied calculus in a former life and am looking for connections with Jacques Derrida's post-structuralist work and calculus (particularly integral calculus). Is the Derrida in the title Jacques Derrida? If so, is the "magic trick" a reference to something in particular, or perhaps a word-play joke? I know that Derrida's work does not tend to be viewed favorably by mathematicians and those in the hard sciences (Sokol and all of that), so I'm also aware it could be referencing that somehow. If anyone knows about the title of this problem, I would greatly appreciate reading what you have to say.

P D - 4 years, 7 months ago

Hello, unfortunately for you, not Jacques, but Bernard. It's a trick used in solving some hard statistical mechanics problems. https://en.m.wikipedia.org/wiki/Bernard_Derrida

Josh Silverman Staff - 4 years, 7 months ago

Thank you very much for the answer and link. I thought that might be the case, though I hadn't come across Bernard Derrida in my search yet because, for some reason, a simple search of "Derrida calculus" first produces results on Jaques where "calculus" is used as a metaphor (either loosely mathematical or medical). Thank you again. This gives me hope that the idea I'm researching hasn't been done yet--or not much.

P D - 4 years, 7 months ago

@Josh Silverman Sir, This is simply a Frullani Integral............so, why the name Derrida??? Just curious...... :)

Aaghaz Mahajan - 2 years, 7 months ago

@Aaghaz Mahajan – It probably has several names, independently discovered/utilized. Derrida was a physicist who used it to solve problems with the partition functions of statistical mechanical problems.

Josh Silverman Staff - 2 years, 7 months ago

@Josh Silverman – Ohhh I see........thanks Sir!!! There is ALWAYS something new to learn from this site............!!!

Aaghaz Mahajan - 2 years, 7 months ago

Mohanish Gaikwad
Mar 4, 2015

If we separate both abs put zt=some other variable say",m".... And integrate with respect to m... The integral itself become zero!! Is z here constant or variable?

same here..Y is that

incredible mind - 6 years, 3 months ago

Is it correct to assume this scenario as a multi variable integral? Only then, I suppose, will the question be really correct.

Sarthak Baghel - 2 years, 7 months ago

Anh Vũ
Mar 5, 2015

For more knowledge about differentiation under the integral sign

Differentiation under the Integral Sign Tutorial, https://www.youtube.com/watch?v=AWA-1rsSSh4 (have more detail solution)

Differentiating under the integral sign, http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf

it s a Frullani's Integral

Mohamed Khalid - 1 year, 3 months ago

Derrida's magic trick

3 solutions

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