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Finding an inverse Laplace transform of an integral that involves the $maxleft0,dotsright$ function

1

1

$begingroup$

I'm trying to tackle a complicated real world (electronics) question. In order to get the last part of the proof I need to find the following 'difficult' integral:

$$mathcalL_texts^-1left[int_textT^frac12xleft(maxlefttextncdotsinleft(2picdot xcdot t-fracpi2right)rightright)cdotexpleft(-textscdot tright)spacetextdtright]_left(t-fractextm2xright)tag1$$

Where $x$, $textz$, $textT$ and $textn$ are all real and positive numbers and $textm$ are the whole positive numbers (integers, including zero).

And I also know that:

$$frac14xletextT<frac12xtag2$$

And:

$$textnspace>spacetextztag3$$

---

My work:

I know that:

$$sinleft(alpha-fracpi2right)=-cosleft(alpharight)tag4$$

So, I can rewrite equation $(1)$ as follows:

$$mathcalL_texts^-1left[int_textT^frac12xleft(maxleft0,leftright)cdotexpleft(-textscdot tright)spacetextdtright]_left(t-fractextm2xright)=$$
$$mathcalL_texts^-1left[int_textT^frac12xleft(maxlefttextncdotcosleft(2picdot xcdot tright)rightright)cdotexpleft(-textscdot tright)spacetextdtright]_left(t-fractextm2xright)tag5$$

And from now on I do not know how to tackle this problem any further. I appreciate any help that you can given me.

calculus trigonometry definite-integrals exponential-function laplace-transform

share|cite|improve this question

edited Mar 19 at 20:37

Jan

asked Mar 19 at 20:28

JanJan

22.1k31440

$endgroup$

add a comment | 

1

1

$begingroup$

I'm trying to tackle a complicated real world (electronics) question. In order to get the last part of the proof I need to find the following 'difficult' integral:

$$mathcalL_texts^-1left[int_textT^frac12xleft(maxlefttextncdotsinleft(2picdot xcdot t-fracpi2right)rightright)cdotexpleft(-textscdot tright)spacetextdtright]_left(t-fractextm2xright)tag1$$

Where $x$, $textz$, $textT$ and $textn$ are all real and positive numbers and $textm$ are the whole positive numbers (integers, including zero).

And I also know that:

$$frac14xletextT<frac12xtag2$$

And:

$$textnspace>spacetextztag3$$

---

My work:

I know that:

$$sinleft(alpha-fracpi2right)=-cosleft(alpharight)tag4$$

So, I can rewrite equation $(1)$ as follows:

$$mathcalL_texts^-1left[int_textT^frac12xleft(maxleft0,leftright)cdotexpleft(-textscdot tright)spacetextdtright]_left(t-fractextm2xright)=$$
$$mathcalL_texts^-1left[int_textT^frac12xleft(maxlefttextncdotcosleft(2picdot xcdot tright)rightright)cdotexpleft(-textscdot tright)spacetextdtright]_left(t-fractextm2xright)tag5$$

And from now on I do not know how to tackle this problem any further. I appreciate any help that you can given me.

calculus trigonometry definite-integrals exponential-function laplace-transform

share|cite|improve this question

edited Mar 19 at 20:37

Jan

asked Mar 19 at 20:28

JanJan

22.1k31440

$endgroup$

add a comment | 

$begingroup$

I'm trying to tackle a complicated real world (electronics) question. In order to get the last part of the proof I need to find the following 'difficult' integral:

$$mathcalL_texts^-1left[int_textT^frac12xleft(maxlefttextncdotsinleft(2picdot xcdot t-fracpi2right)rightright)cdotexpleft(-textscdot tright)spacetextdtright]_left(t-fractextm2xright)tag1$$

Where $x$, $textz$, $textT$ and $textn$ are all real and positive numbers and $textm$ are the whole positive numbers (integers, including zero).

And I also know that:

$$frac14xletextT<frac12xtag2$$

And:

$$textnspace>spacetextztag3$$

---

My work:

I know that:

$$sinleft(alpha-fracpi2right)=-cosleft(alpharight)tag4$$

So, I can rewrite equation $(1)$ as follows:

$$mathcalL_texts^-1left[int_textT^frac12xleft(maxleft0,leftright)cdotexpleft(-textscdot tright)spacetextdtright]_left(t-fractextm2xright)=$$
$$mathcalL_texts^-1left[int_textT^frac12xleft(maxlefttextncdotcosleft(2picdot xcdot tright)rightright)cdotexpleft(-textscdot tright)spacetextdtright]_left(t-fractextm2xright)tag5$$

And from now on I do not know how to tackle this problem any further. I appreciate any help that you can given me.

calculus trigonometry definite-integrals exponential-function laplace-transform

share|cite|improve this question

edited Mar 19 at 20:37

Jan

asked Mar 19 at 20:28

JanJan

22.1k31440

$endgroup$

share|cite|improve this question

edited Mar 19 at 20:37

Jan

asked Mar 19 at 20:28

JanJan

22.1k31440

share|cite|improve this question

edited Mar 19 at 20:37

Jan

asked Mar 19 at 20:28

JanJan

22.1k31440

asked Mar 19 at 20:28

JanJan

22.1k31440

asked Mar 19 at 20:28

JanJan

22.1k31440