Toronto function

In mathematics, the Toronto function T(m,n,r) is a modification of the confluent hypergeometric function defined by Heatley (1943), Weisstein, as

T ( m , n , r ) = r 2 n − m + 1 e − r 2 Γ ( 1 2 m + 1 2 ) Γ ( n + 1 ) 1 F 1 ( 1 2 m + 1 2 ; n + 1 ; r 2 ) . {\displaystyle T(m,n,r)=r^{2n-m+1}e^{-r^{2}}{\frac {\Gamma ({\frac {1}{2}}m+{\frac {1}{2}})}{\Gamma (n+1)}}{}_{1}F_{1}({\textstyle {\frac {1}{2}}}m+{\textstyle {\frac {1}{2}}};n+1;r^{2}).}
Later, Heatley (1964) recomputed to 12 decimals the table of the M(R)-function, and gave some corrections of the original tables. The table was also extended from x = 4 to x = 16 (Heatley, 1965). An example of the Toronto function has appeared in a study on the theory of turbulence (Heatley, 1965).

References

  • Heatley, A. H. (1943), "A short table of the Toronto function", Trans. Roy. Soc. Canada Sect. III., 37: 13–29, MR 0010055
  • Heatley, A. H. (1964), "A short table of the Toronto function", Mathematics of Computation, 18, No.88: 361
  • Heatley, A. H. (1965), "An extension of the table of the Toronto function", Mathematics of Computation, 19, No.89: 118-123
  • Weisstein, E. W., "Toronto Function", From Math World - A Wolfram Web Resource