content_image/350825.jpg content_image/350824.jpg content_image/350823.jpg content_image/350822.jpg content_image/350821.jpg content_image/350820.jpg content_image/350819.jpg content_image/350818.jpg content_image/350817.jpg content_image/350816.jpg
Take a change z = r$_{+}$ / r , Eqs. (3.4) and (3.5) can be rewritten as 

<img src='content_image/350825.jpg'>

<img src='content_image/350824.jpg'>

where the prime denotes di ff erentiation with z . Regularity of the functions at the horizon z = 1 requires 

And near the boundary z = 0 we have 

<img src='content_image/350823.jpg'>

<img src='content_image/350822.jpg'>

We will set $^{C}$− = 0 and fix ρ in the following discussion. With the help of the regular horizon boundary condition (3.13), the leading order approximate solutions near the horizon, z = 1, for the Eqs. (3.11) and (3.12) can be expressed as 

where a ≡ ψ (1) and b ≡ − φ ′ (1) with a , b > 0 which makes ψ ( z ) and φ ( z ) positive near the horizon. Matching smoothly the solutions (3.15), (3.16) with (3.14) at an intermediate point z$_{m}$ with 0 < z$_{m}$ < 1, we have 

with 

<img src='content_image/350821.jpg'>

<img src='content_image/350820.jpg'>

<img src='content_image/350819.jpg'>

<img src='content_image/350818.jpg'>

<img src='content_image/350817.jpg'>

<img src='content_image/350816.jpg'>

