Is the swimming pool water harmful to frogs who jump in? If so, how?
The chlorine concentration in pools is +- 0,5 mg/L.


+


  0.002 mg/L will fatally damage the sensitive skin on tadpoles, frogs, salamanders and other amphibians.


another source:


  Free chlorine (Cl2) is a greenish gas that is well known for its
  highly toxic properties as can be attested to by the thousands of
  soldiers that died and were severely injured from chlorine exposure
  during World War I. In water, chlorine is the most toxic substance
  that we will discuss. Ironically for the amphibian keeper, it is this
  toxic nature of chlorine and its ability to denature proteins, which
  makes its encounter inevitable. Chlorine is generally used as an
  antibacterial agent in municipal water supplies and may be present in
  concentrations of over 9 mg/1 in some tap water (measured in Houston,
  Texas as an example, although levels were generally lower). The
  concentration of chlorine in municipal water supplies can vary greatly
  from day to day, or even hour to hour, depending on conditions at the
  water treatment facilities. Concentrations as low as 0.0034 mg/1 have
  been noted to reduce reproduction in fathead minnows with 72 hour
  LC100 (lethal concentration for 100% kill) at 0.15 mg/1 (Arthur and
  Eaton,1971). LC50 (96 hour) for the shiners (Notemigonus chrysoleucas)
  was as low as 0.19 mg/1 (Esvelt et al., 1971). The concentrations
  found in municipal water supplies are many times greater than the
  minimum lethal concentrations for many aquatic life forms.


Thus, the well being of the frog in the water of the swimming pool depends on the time the frog spends in the water. Eventually, with extended exposure the chlorine concentration will exceed the one compatible will life and the frog will die. This time is multifactorial variable, thus cannot be assessed easily - the weight and the skin surface area and skin permeability will affect the frog survival in a great degree.