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In writing a control proof, I need to type "k times" below the two products as shown in the code below. How can I do it?

\documentclass[12pt]{article}

\usepackage{amsmath}


\begin{document}


To prove: $\boxed{  [ \Phi(t) ]^k = \Phi(k t). }$


    Since $\Phi(t) = e^{A t}$, we find that
    $$
    [ \Phi(t) ]^k = \Phi(t) \Phi(t) \cdots \Phi(t) = e^{A t}  e^{A t} \cdot e^{A t} = e^{k A t}
    $$




\end{document}

LaTeX Output

Peter Mortensen
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Dr. Sundar
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  • I fully agree that it is "generally understood" when we write a research paper and we don't need to spell out "k times" with "underbrace".. However, for lecture notes, I like to add this for "pedagogy".. I am writing lecture notes for a control course that I teach to M. Tech engineering students, and I like to ensure that they understand in detail. Thanks. – Dr. Sundar Apr 26 '22 at 05:04

1 Answers1

10

You can use the \underbrace macro.

\documentclass[12pt]{article}

\usepackage{amsmath}


\begin{document}


To prove: $\boxed{  [ \Phi(t) ]^k = \Phi(k t). }$


Since $\Phi(t) = e^{A t}$, we find that
\[ 
    [ \Phi(t) ]^k
    = \underbrace{ \Phi(t) \Phi(t) \cdots \Phi(t) }_{k \text{ times}}
    = \underbrace{ e^{A t}  e^{A t} \cdots e^{A t} \mathstrut }_{k \text{ times}}
    = e^{k A t}
\]




\end{document}

output

A \mathstrut is placed to improve the spacing of the second brace.

Note that you should use \[...\] in LaTeX instead of the deprecated $$...$$. See this post for more information.

plante
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    Amazing! It works very well. I will accept your answer after a day or two as moderators advised me to wait for other answers as well! Thanks a lot! – Dr. Sundar Apr 25 '22 at 08:40
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    I am very comfortable using $$ .... $$ - I still use LaTeX 2e only.. I had no issues so far in MikTeX - thanks! – Dr. Sundar Apr 25 '22 at 08:41
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    plante: I assume there is a similar "\overbrace" command in amsmath that works in a similar way.. I shall test it also, thanks! – Dr. Sundar Apr 25 '22 at 08:43
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    plante: "\overbrace" also works well.. I guess we use "\underbrace" more commonly for derivations like this, thanks! $$ [ \Phi(t) ]^k = \overbrace{ \Phi(t) \Phi(t) \cdots \Phi(t) }^{k \text{ times}}= \overbrace{e^{A t} e^{A t} \cdot e^{A t} }^{k \ \text{times}} = e^{k A t} = \Phi(k t) $$ – Dr. Sundar Apr 25 '22 at 08:45
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    @Dr.Sundar you should not be comfortable with $$, it is wrong and has never been supported LaTeX syntax, it is not even mentioned in the official LaTeX book. This has nothing to do with LaTeX2e/laTeX3, $$ was not supported in LaTeX2.09 either. – David Carlisle Apr 25 '22 at 08:48
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    David: OK, I will start typing in \[...\] for display equations from now on.. – Dr. Sundar Apr 25 '22 at 08:50
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    Note that \underbrace isn't from amsmath. It's part of the LaTeX (and plain TeX) kernel. – campa Apr 25 '22 at 09:13
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    campa: thanks for reminding, I have corrected the answer – plante Apr 25 '22 at 09:14