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## Main Question or Discussion Point

I have a quick question about integration after a change of variables has been made.

Suppose there is a function [tex] R(t_{1},t_{2}) [/tex] that actually just

depends on the difference [tex] t_{1} - t_{2} [/tex]. The goal is then to

simplify the following integral:

[tex]

\frac{1}{T^{2}}\int^{T}_{0}\int^{T}_{0} R(t_{1},t_{2}) dt_{1}dt_{2}

[/tex]

by using the substitution [tex] t_{1}' = t_{1} [/tex] and [tex] t_{2}'= t_{1} - \tau [/tex].

A straight substitution yields:

[tex]

\frac{1}{T^{2}}\int \int^{T}_{0} R(\tau) dt_{1}'(dt_{1}' - d\tau)

[/tex]

I am uncertain about two things:

1) the integration bounds on the outer integral after the substitution has been made

2) whether or not [tex]dt_{1}'[/tex] in the outer integral is zero since

[tex]dt_{1}[/tex] is held constant when integrating over [tex]dt_{2}[/tex] before the substitution was made.

As a heads up the final result is supposed to be:

[tex]

\frac{1}{T^{2}}\int^{T}_{-T}\left(T-\left|\tau\right|\right)R(\tau) d\tau

[/tex]

Thanks in advance for any responses.

Suppose there is a function [tex] R(t_{1},t_{2}) [/tex] that actually just

depends on the difference [tex] t_{1} - t_{2} [/tex]. The goal is then to

simplify the following integral:

[tex]

\frac{1}{T^{2}}\int^{T}_{0}\int^{T}_{0} R(t_{1},t_{2}) dt_{1}dt_{2}

[/tex]

by using the substitution [tex] t_{1}' = t_{1} [/tex] and [tex] t_{2}'= t_{1} - \tau [/tex].

A straight substitution yields:

[tex]

\frac{1}{T^{2}}\int \int^{T}_{0} R(\tau) dt_{1}'(dt_{1}' - d\tau)

[/tex]

I am uncertain about two things:

1) the integration bounds on the outer integral after the substitution has been made

2) whether or not [tex]dt_{1}'[/tex] in the outer integral is zero since

[tex]dt_{1}[/tex] is held constant when integrating over [tex]dt_{2}[/tex] before the substitution was made.

As a heads up the final result is supposed to be:

[tex]

\frac{1}{T^{2}}\int^{T}_{-T}\left(T-\left|\tau\right|\right)R(\tau) d\tau

[/tex]

Thanks in advance for any responses.