Eikonal Blog

2010.02.15

Lutzky’s expansion

Filed under: mathematics — Tags: — sandokan65 @ 22:16

\frac{d e^{z}}{dt} = e^z \left\{\frac{dz}{dt},\frac{e^z-1}{z}\right\}

e^{z(t)} :\equiv e^{x}e^{t y}

  • z(t) = \sum_{l=0}^\infty z_l t^l
  • z_0 = x
  • z_1 = \left\{ y , \frac{x}{e^x-1}\right\} = y - \frac1 [y,x] + \frac1{12} \{y,x^2\} - \frac1{720} \{y,x^4\} + \cdots

Here the expansion \frac{x}{e^x-1} = \sum_{k=0}^\infty \frac{B_k}{k!} x^k was used.

Calculating z_2

Definition:

  • \frac{x e^{\xi x}}{e^x-1} = \sum_{k=0}^\infty \varphi_k(\xi) x^k
  • \frac{e^{\xi x} -1}{e^x-1} = \sum_{k=0}^\infty \Psi_k(\xi) x^k
  • \varphi_k(\xi) = \frac1{k!} \sum_{m=0}^\infty \binom{k}{m} B_m \xi^{k-m}
  • \partial_\xi \varphi_{k+1}(\xi) = \varphi_k(\xi)
  • \varphi_k(0)= \frac1{k!}B_k,
  • \varphi_k(1)= \varphi_k(0) (for k\ne 1),
  • \varphi_0(\xi) \equiv 1
  • \int_0^1 \varphi_k(\xi) d\xi = \delta_{k,0},
  • \Psi_{k}(\xi) = \varphi_{k+1}(\xi) - \varphi_{k+1}(0),
  • \int_0^1 \varphi_k(\xi) \Psi_{l}(\xi)  d\xi = \delta_{k,0}\varphi_{l+1}(0) \theta(l) + \delta_{l,0}\varphi_{k+1}(0) \theta(k),

Now:
z_2 = - \sum_{l=1}^\infty \sum_{k=0}^\infty \{y,x^l,y,x^k\} \frac{B_{l+1}B_k}{(l+1)!k!} -\frac12 \sum_{k=1}^\infty \sum_{l=1}^\infty \sum_{j=0}^\infty I_{kl} \{\Delta_{kl}, x^j\}  \frac{B_j}{j!}
where:

  • \{y,x^k,y\} :\equiv [\{y,x^k\},y],
  • \{y,x^l,y,x^k\} :\equiv \{\{y,x^l,y\},x^k\},
  • \Delta_{kl} :\equiv [\{y,x^l,\{y,x^k\}] = - \Delta_{lk},
  • I_{kl} :\equiv \int_0^1 \varphi_k(\xi) \varphi_{l+1}(\xi), (I_{l+2,l}=0)

The more compact expression is:
z_2 = -\frac12 \left\{ J, \frac{x}{e^x-1}\right\}
where
J :\equiv \int_0^1 d\xi \left[\{z_1,e^{x\xi}\}, \left\{z_1,\frac{e^{x\xi}-1}{x}\right\}\right] = \int_0^1 d\xi \left[\{y,\frac{xe^{x\xi}}{e^x-1}\}, \left\{y,\frac{e^{x\xi}-1}{e^{x}-1}\right\}\right].

The Hausdorff recursion formula:
z_n = \frac1{n} \left(\left\{y, \frac{x}{e^x-1}\right\} D_x \right) z_{n-1}.


Source: T1265 = M. Lutzky “Parameter Differentiation of Exponential Operators and the Baker-Campbell-Hausdorff Formula”; Journal of Mathematics Physics, Vol 9, N 7, July 1968.

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