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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

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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, 2022 at 5:04

1 Answer 1

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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.

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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, 2022 at 8:40
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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, 2022 at 8:41
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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. Apr 25, 2022 at 8:48
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    Note that \underbrace isn't from amsmath. It's part of the LaTeX (and plain TeX) kernel.
    – campa
    Apr 25, 2022 at 9:13
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    campa: thanks for reminding, I have corrected the answer
    – plante
    Apr 25, 2022 at 9:14

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