1.6.3 Matrix-Based Infinite Integration and Differentiation

The pre­vi­ous sec­tion was an an­a­lyt­ic treat­ment of in­fi­nite in­te­gra­tion and in­fi­nite dif­fer­en­ti­a­tion, and in this sec­tion, we apply in­te­gra­tion and dif­fer­en­ti­a­tion op­er­a­tors to demon­strate these in­fi­nite func­tions in matrix-based forms. Start­ing with the in­fi­nite in­te­gra­tion

we define the in­te­gra­tion op­er­a­tor ${\psi\left( \theta_{r}^{\left( m,X_{1} \right)} \right)}^{- n}$, for the ${iX}_{1}$- type matrix, as

where the en­tries of the matrix will act by way of a prod­uct with an­oth­er matrix to per­form the in­te­gra­tion op­er­a­tion with re­spect to $\theta_{j}$. We denote the $iX_{1}$ matrix, as

for $\mathfrak{su}\left( \theta_{1}^{\left( 3,X_{1} \right)} \right)^{1}$, a Lie al­ge­bra el­e­ment of unit radial mag­ni­tude, $r = 1$, with the su­per­script $\left( 3,X_{1} \right)$, in­di­cat­ing the $\mathbb{C}^{3}$ di­men­sion­al space, and an $X_{1}$-type matrix. So, it should be noted that $\theta_{r}^{m}$ is de­fined $\in \mathbb{C}$, but in this case we have a unit radial mag­ni­tude, which means that we are work­ing with com­plex num­bers that lie on the com­plex unit sphere, and specif­i­cal­ly, $\theta_{1}^{m} = e^{i\theta}$.

Now, we can ex­press the in­fi­nite in­te­gra­tion func­tion in a matrix-based form, as

where the ex­po­nent value $n$, for the op­er­a­tor ${\psi\left( \theta_{1}^{\left( 3,X_{1} \right)} \right)}^{- n}$, de­ter­mines the in­te­gra­tion order and the de­nom­i­na­tor's value in the matrix, and $\epsilon$ is the iden­ti­ty matrix

Con­tin­u­ing the in­te­gra­tion process, the first-order in­te­gral yields the $\mathfrak{su}\left( \theta_{1}^{\left( 3,X_{1} \right)} \right)^{1}$ in­te­gra­tion matrix of el­e­ments

and for the evolv­ing process, we find the next three Lie al­ge­bras for $\mathfrak{su}\left( \theta_{1}^{\left( 3,X_{1} \right)} \right)^{n}$:

For the ex­pres­sions above, notice that al­ter­nat­ing in­te­gra­tions occur for the $\theta_{2}{,\theta}_{3}$ vari­ables, which is re­lat­ed to whether we have an even or odd ex­po­nent for the op­er­a­tor's $n$. Odd in­te­gra­tion op­er­a­tors will yield el­e­ments of the odd trigono­met­ric func­tions, and even in­te­gra­tion op­er­a­tors will yield el­e­ments of the even trigono­met­ric func­tions. So, this fash­ion of in­fi­nite in­te­gra­tion fits nicely into the con­text of finite group theory, as we are using a mul­ti­pli­ca­tion op­er­a­tion with Lie al­ge­bras to cal­cu­late the matrix ex­po­nen­tial map­ping, for each $\mathfrak{su}\left( \theta_{r}^{m} \right)^{n}$, re­cur­sive­ly re­lat­ed to its $\mathfrak{su}\left( \theta_{r}^{m} \right)^{n - 1}$ pre­de­ces­sor.

Re­gard­ing the in­fi­nite in­te­gra­tion sum­ma­tion of Lie al­ge­bra el­e­ments, de­not­ed as $\mathfrak{SU}\left( \theta_{r}^{m} \right)^{n}$, we define the fol­low­ing:

De­f­i­n­i­tion (1.6.53) – $\mathfrak{SU}\left( \mathbf{\theta}_{\mathbf{r}}^{\left( \mathbf{m} \right)} \right)^{\mathbf{n}}$ is the sum­ma­tion of Lie al­ge­bra el­e­ments $\mathfrak{su}\left( \theta_{r}^{m} \right)^{n}\in \mathfrak{g}$, to order-$n$, form­ing the el­e­ments of the in­fi­nite series of in­te­gra­tion/matrix ex­po­nen­ti­a­tion,

Given in the op­er­a­tor de­f­i­n­i­tion of ${\psi\left( \theta_{r}^{m} \right)}^{- n}$, the neg­a­tive ex­po­nent $- n$, de­notes in­te­gra­tion, and now we con­sid­er anal­o­gous no­ta­tion for in­te­gra­tions of the Lie al­ge­bra el­e­ments $\mathfrak{su}\left( \theta_{r}^{m} \right)^{n},$in the de­f­i­n­i­tion that fol­lows.

De­f­i­n­i­tion (1.6.54) – ${\mathbf{\psi}\left( \mathbf{\theta}_{\mathbf{r}}^{\mathbf{m}} \right)}^{- \mathfrak{n}}$ is the in­fi­nite fac­to­r­i­al in­te­gra­tion op­er­a­tor de­fined, as

which forms prod­ucts with Lie al­ge­bra el­e­ments$\ \mathfrak{su}\left( \theta_{r}^{m} \right)^{n}\in \mathfrak{g}$, with the rec­i­p­ro­cal fac­to­r­i­al el­e­ments $\mathfrak{n}^{\mathbf{k}}$, de­fined by

De­f­i­n­i­tion (1.6.55) – $\mathfrak{SU}\left( \mathbf{\theta}_{\mathbf{r}}^{\mathbf{m}} \right)^{\mathbf{-}\mathfrak{n}^{\mathbf{k}}}$ is the sum­ma­tion of the Lie al­ge­bra el­e­ments $\mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{j}}}\in \mathfrak{g}$, to fac­to­r­i­al and ex­po­nent order-k, that form the el­e­ments of the in­fi­nite series of in­te­gra­tion/matrix ex­po­nen­ti­a­tion

Using the in­fi­nite fac­to­r­i­al in­te­gra­tion op­er­a­tor, we can ex­press the Lie al­ge­bra el­e­ments $\mathfrak{su}\left( \theta_{r}^{m} \right)^{n}$, as fol­lows:

re­call­ing that the $\left( \mathfrak{n}^{\mathbf{k}} \right)^{- \mathbf{1}}\mathbf{=}\frac{\mathbf{1}}{k\mathbf{!}}$ in the ex­pres­sions above, shows the equiv­a­lence re­la­tion­ship $\mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{k}}}= \mathfrak{su}\left( \theta_{r}^{m} \right)^{k}.$

With the fac­to­r­i­al in­te­gra­tion op­er­a­tor de­fined, we can now con­sid­er an it­er­a­tive means to cal­cu­late any series of Lie al­ge­bra el­e­ments $\mathfrak{SU}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{k}}}$, by using in­duc­tion and the static fac­to­r­i­al in­te­gra­tion op­er­a­tor ${\psi\left( \theta_{r}^{m} \right)}^{- \mathfrak{n}}$, de­scribed in Lemma (1.6.60) and The­o­rem (1.6.61).

Lemma (1.6.60) – The fol­low­ing re­cur­sive re­la­tion­ship holds

Proof – We start by taking the in­verse of ${\psi\left( \theta_{r}^{m} \right)}^{- \mathfrak{n}}$, then mul­ti­ply it to both sides

and

$\square$

Lemma (1.6.61) – The fol­low­ing re­cur­sive re­la­tion­ship holds

Proof – We start by taking the in­verse of ${\psi\left( \theta_{r}^{m} \right)}^{- \mathfrak{n}}$, and mul­ti­ply it to both sides

and

All $\mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{2}\mathcal{k} +\mathbf{1}}}$ are odd, and have the form

where­as all $\mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{2}\mathcal{k}}}$ are even, having the form

The fol­low­ing holds for all odd $\mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{2}\mathcal{k} +\mathbf{1}}}$, and even $\mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{2}\mathcal{k}}}$

$\square$

The­o­rem (1.6.62) – The sum­ma­tion of the Lie al­ge­bra el­e­ments $\mathfrak{SU}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{k} + \mathbf{1}}}$to the $(k + 1)^{th}$ order that form the el­e­ments of the in­fi­nite series of in­te­gra­tion/matrix ex­po­nen­ti­a­tion, is re­cur­sive­ly re­lat­ed as a prod­uct of the in­fi­nite in­te­gra­tion op­er­a­tor ${\psi\left( \theta_{r}^{m} \right)}^{- \mathfrak{n}} \times$ $\mathfrak{SU}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}}$, and the iden­ti­ty el­e­ment $\epsilon$, as

Proof – We first show that this holds for $\mathfrak{SU}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{2}}},$ where­by de­f­i­n­i­tion we have

there­fore,

Now, by in­duc­tion

ex­pand­ing the sum­ma­tion, $\left( \sum_{\mathbf{j} = 0}^{\mathbf{k}}{\ \mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{j}}}} \right)$,

and know­ing the re­cur­sive re­la­tion­ship, $\mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{k} + \mathbf{1}}} = {\psi\left( \theta_{r}^{m} \right)}^{- \mathfrak{n}}\times \mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{k}}}$, and $\epsilon = \mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{0}}}$, the ex­pres­sion be­comes

By De­f­i­n­i­tion (1.6.55),

and there­fore, it holds that

$\square$

Dif­fer­en­ti­a­tion is de­fined as the in­verse of in­te­gra­tion, which leads us to define the in­fi­nite dif­fer­en­ti­a­tion op­er­a­tor ${\psi\left( \theta_{r}^{m} \right)}^{\mathfrak{n}}$, as the in­verse of the in­fi­nite in­te­gra­tion op­er­a­tor ${\psi\left( \theta_{r}^{m} \right)}^{- n}$.

De­f­i­n­i­tion (1.6.63) – ${\mathbf{\psi}\left( \mathbf{\theta}_{\mathbf{r}}^{\mathbf{m}} \right)}^{\mathfrak{n}}$ is the in­fi­nite fac­to­r­i­al dif­fer­en­tial op­er­a­tor, for a pos­i­tive-valued ex­po­nent $\mathfrak{n}$, and is de­fined as the in­verse of the in­fi­nite fac­to­r­i­al in­te­gra­tion op­er­a­tor

for this in­verse re­la­tion­ship, de­fined as

Lemma (1.6.64) – The fol­low­ing re­cur­sive re­la­tion­ship holds for dif­fer­en­ti­a­tion

Proof – As de­fined,$\mathbf{\ }{\psi\left( \theta_{r}^{m} \right)}^{\mathfrak{n}}$ is the in­verse of ${\psi\left( \theta_{r}^{m} \right)}^{- \mathfrak{n}},$and we can mul­ti­ply both sides by ${\psi\left( \theta_{r}^{m} \right)}^{- \mathfrak{n}}$

*and this is equiv­a­lent to Lemma (1.6.54), which proved the in­te­gra­tion re­cur­sive re­la­tion­ship that*

$\square$

Con­sid­er­ing in­fi­nite dif­fer­en­ti­a­tion's in­verse re­la­tion­ship to in­fi­nite in­te­gra­tion, we state and prove the the­o­rem:

The­o­rem (1.6.65) – The sum­ma­tion of the Lie al­ge­bra el­e­ments $\mathfrak{SU}\left( \theta_{r}^{m} \right)^{\mathfrak{n}^{\mathbf{k}}}$, to order-$\mathbf{k}$, form­ing the el­e­ments of the in­fi­nite dif­fer­en­tial series, is in­verse­ly re­lat­ed to in­fi­nite in­te­gra­tion func­tion

Proof – We first show that this holds for $\mathfrak{SU}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{2}}},$ where­by the de­f­i­n­i­tion, we have that

there­fore,

By in­duc­tion

ex­pand­ing the sum­ma­tion $\left( \sum_{\mathbf{j} = 0}^{\mathbf{k} + \mathbf{1}}{\ \mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{j}}}} \right),$ yields

where know­ing the re­cur­sive re­la­tion­ship, $\mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{k} + \mathbf{1}}} = {\psi\left( \theta_{r}^{m} \right)}^{- \mathfrak{n}}\times \mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{k}}}$, and that the in­fi­nite dif­fer­en­tial op­er­a­tor is the in­verse of ${\psi\left( \theta_{r}^{m} \right)}^{- \mathfrak{n}}$, we have that

Since $\left( {\psi\left( \theta_{r}^{m} \right)}^{\mathfrak{n}} \times {\psi\left( \theta_{r}^{m} \right)}^{- \mathfrak{n}} \right) = \epsilon$,

As $\mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{0}}} = \epsilon$, the sum­ma­tion be­comes

and by the in­verse re­la­tion­ship ${\psi\left( \theta_{r}^{m} \right)}^{\mathfrak{n}}\times \mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{k} + \mathbf{1}}}= \mathfrak{su}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{k}}},$ we have

which is the same as De­f­i­n­i­tion (1.6.55)

which proves that

$\square$

The ad­van­tage of using the op­er­a­tor no­ta­tion, with pos­i­tive and neg­a­tive ex­po­nents, as $\mathfrak{n}$, is when using prod­ucts of op­er­a­tors and ma­tri­ces, the ex­po­nents are simply a prod­uct of the ex­po­nents, as seen in the ex­am­ples

given that, $\mathfrak{n}^{\mathbf{1}}\left( \frac{\mathbf{1}}{\mathfrak{n}^{\mathbf{k} + \mathbf{1}}} \right)\mathbf{=}{\mathbf{1}/\mathfrak{n}}^{\mathbf{k}}$, and

as, $\left( \frac{\mathbf{1}}{\mathfrak{n}^{\mathbf{1}}} \right)\left( \frac{\mathbf{1}}{\mathfrak{n}^{\mathbf{k}}} \right)\mathbf{=}\frac{\mathbf{1}}{\mathfrak{n}^{\mathbf{k} + \mathbf{1}}}\mathbf{.}$

For the other ${iX}_{j}$-type Lie al­ge­bras, it is straight­for­ward to define in­fi­nite fac­to­r­i­al in­te­gra­tion and dif­fer­en­tial op­er­a­tors as fol­lows:

The ex­po­nen­tial map­ping is forth­right in terms of our in­fi­nite in­te­gra­tion se­quence, where

and using, once again, the ${iX}_{1}$-type matrix, the above ex­pands to the fol­low­ing matrix series form

From the ex­po­nen­tial map­ping, we can trans­form back to the Lie al­ge­bra $\mathfrak{g}$, by way of the fol­low­ing re­la­tion

We can now define the matrix log­a­rithm func­tion in terms of the in­fi­nite dif­fer­en­tial, as the log­a­rithm of the series

which can be ex­pressed in terms of op­er­a­tor notion

where suc­ces­sive dif­fer­en­tial it­er­a­tions are taken, start­ing at $k = \infty$, pro­ceed­ing all the way to $k = 2$, for the prod­uct ${\psi\left( \theta_{r_{j}}^{m} \right)}^{\mathfrak{n}}\left( \mathfrak{SU}\left( \theta_{r}^{m} \right)^{- \mathfrak{n}^{\mathbf{k}}} - \epsilon \right)$. If the radial com­po­nent is not as­sumed to be uni­tary, we find that the sub­tan­gent form $\ln\left( \theta_{r}^{m} \right)$, is equal to

and for $\ln{(r) \neq 0}$, that the matrix en­tries in the in­te­gra­tion and dif­fer­en­ti­a­tion op­er­a­tors take the fol­low­ing forms, re­spec­tive­ly,

Given this, the non-uni­tary in­fi­nite in­te­gra­tion and dif­fer­en­ti­a­tion op­er­a­tors can be rep­re­sent­ed, as