Introduction To Hyperbolic Functions Pdf 20

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Since the hyperbolic planform is a hyperbolic extension, a smooth monotonic weight function W will necessarily be a monotonically decreasing function of r. The hyperbolic planform may be written down as,

If we consider Eqs. (22) and (23), then the case under consideration is a variant of a problem discussed in [6] in the context of the nonlinear wave equation, or our work on periodic solutions . We give the solution first. Given a monotonically decreasing weight function W, the general solution may be written down as,

To show this we first remark that the hyperbolic solution (23) obeys the same equations as a planar wave solution that is invariant under the action of the same isometry. If we consider the case when a subgroup of U(1, 1) leaves the solution invariant, then Eq. (23) is a wave equation on a 2-dimensional space with a source term that depends upon the r coordinate.

Let W(r) denote the integral over the solid angle in the Poincar disk, and let W’(r) denote the derivative of W(r), i.e. the integral of the radial derivative of W(r) over r. Since W(r) is necessarily a monotonically decreasing function of r, we have W’(r) < 0. The integral over the disk can be evaluated as:

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