The Spiral of Theodorus, Numerical Analysis, and Special Functions

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1 Theo p. / The Spiral of Theodorus, Numerical Analysis, and Special Functions Walter Gautschi Purdue University

2 Theo p. 2/ Theodorus of ca B.C.

3 Theo p. 3/ spiral of Theodorus

4 Theo p. 3/ spiral of Theodorus numerical analysis k 3/2 + k /2

5 Theo p. 3/ spiral of Theodorus numerical analysis k 3/2 + k /2 =

6 Theo p. 3/ spiral of Theodorus numerical analysis k 3/2 + k /2 = special functions F (x) = e x2 x e t2 dt Dawson s integral

7 Theo p. 4/ discrete spiral of Theodorus (also known as Quadratwurzelschnecke ; Hlawka, 98) 2 T 5 T 4 T 3 T T T 2 T 8 T 7 T n 2 Tn

8 Theo p. 4/ discrete spiral of Theodorus (also known as Quadratwurzelschnecke ; Hlawka, 98) 2 T 5 T 4 T 3 T T T 2 T 8 T 7 T n 2 Tn parametric representation T (α) C, α

9 Theo p. 4/ discrete spiral of Theodorus (also known as Quadratwurzelschnecke ; Hlawka, 98) 2 T 5 T 4 T 3 T T T 2 T 8 T 7 T n 2 Tn parametric representation T (α) C, α defining properties T (n) = T n T n = n T n+ T n = n =,, 2,...

10 Theo p. 5/ problem: Interpolate the discrete Theodorus spiral by a smooth (or even analytic) spiral

11 Theo p. 5/ problem: Interpolate the discrete Theodorus spiral by a smooth (or even analytic) spiral solution (P.J. Davis, 993; inspired by Euler s work on the gamma function)

12 Theo p. 5/ problem: Interpolate the discrete Theodorus spiral by a smooth (or even analytic) spiral solution (P.J. Davis, 993; inspired by Euler s work on the gamma function) T (α) = + i/ k + i/ k + α, α (i = )

13 Theo p. 5/ problem: Interpolate the discrete Theodorus spiral by a smooth (or even analytic) spiral solution (P.J. Davis, 993; inspired by Euler s work on the gamma function) T (α) = + i/ k + i/ k + α, α (i = ) properties T (α) = α ( ) T (α + ) = + i α T (α)

14 Theo p. 5/ problem: Interpolate the discrete Theodorus spiral by a smooth (or even analytic) spiral solution (P.J. Davis, 993; inspired by Euler s work on the gamma function) T (α) = + i/ k + i/ k + α, α (i = ) properties T (α) = α ( ) T (α + ) = + i α T (α) heart of the spiral: T (α), α 2

15 Theo p. 5/ problem: Interpolate the discrete Theodorus spiral by a smooth (or even analytic) spiral solution (P.J. Davis, 993; inspired by Euler s work on the gamma function) T (α) = + i/ k + i/ k + α, α (i = ) properties T (α) = α ( ) T (α + ) = + i α T (α) heart of the spiral: T (α), α 2 (Gronau, 24) Davis s function is the unique solution of the above difference equation with T (α) and arg T (α) monotonically increasing, and T () =.

16 a little bit of number theory Theo p. 6/

17 Theo p. 6/ a little bit of number theory distribution of the angles ϕ n = T T T n+ = n sin k +, n =, 2, 3,...

18 Theo p. 6/ a little bit of number theory distribution of the angles ϕ n = T T T n+ = n sin k +, n =, 2, 3,... equidistribution (Hlawka, 979) The sequence {ϕ n } n= is equidistributed mod 2π

19 Theo p. 6/ a little bit of number theory distribution of the angles ϕ n = T T T n+ = n sin k +, n =, 2, 3,... equidistribution (Hlawka, 979) The sequence {ϕ n } n= is equidistributed mod 2π more generally ϕ n (α) = T (α)t T (α + n) = n sin k + α < α < 2, n =, 2, 3,...

20 Theo p. 6/ a little bit of number theory distribution of the angles ϕ n = T T T n+ = n sin k +, n =, 2, 3,... equidistribution (Hlawka, 979) The sequence {ϕ n } n= is equidistributed mod 2π more generally ϕ n (α) = T (α)t T (α + n) = n sin k + α < α < 2, n =, 2, 3,... The sequence {ϕ n (α)} n= is equidistributed mod 2π for any α with < α < 2 (Niederreiter, Feb. 3, 29)

21 Theo p. 7/ logarithmic derivative of T (α) T (α) T (α) = + i/ k + α + i/ k d dα ( + i/ ) k + i/ k + α

22 Theo p. 7/ logarithmic derivative of T (α) T (α) T (α) = + i/ k + α + i/ k d dα ( + i/ ) k + i/ k + α = ( + i/ k + α ) i 2 (k + α ) 3/2 ( + i/ k + α ) 2

23 Theo p. 7/ logarithmic derivative of T (α) T (α) T (α) = + i/ k + α + i/ k d dα ( + i/ ) k + i/ k + α = ( + i/ k + α ) i 2 (k + α ) 3/2 ( + i/ k + α ) 2 = i 2 (k + α )( k + α + i)

24 Theo p. 7/ logarithmic derivative of T (α) T (α) T (α) = + i/ k + α + i/ k d dα ( + i/ ) k + i/ k + α = ( + i/ k + α ) i 2 (k + α ) 3/2 ( + i/ k + α ) 2 = i 2 = i 2 (k + α )( k + α + i) k + α i (k + α )(k + α)

25 Theo p. 8/ logarithmic derivative of T (α) (cont ) = 2 (k + α )(k + α) + i 2 (k + α ) 3/2 + (k + α ) /2

26 Theo p. 8/ logarithmic derivative of T (α) (cont ) = 2 (k + α )(k + α) + i 2 = 2 (k + α ) 3/2 + (k + α ) /2 ( k + α ) + i k + α 2 U(α)

27 Theo p. 8/ logarithmic derivative of T (α) (cont ) = 2 (k + α )(k + α) + i 2 where = 2 (k + α ) 3/2 + (k + α ) /2 ( k + α ) + i k + α 2 U(α) = 2α + i 2 U(α) U(α) = (k + α ) 3/2 + (k + α ) /2

28 T (α) T (α) = 2α + i 2 U(α) Theo p. 9/

29 Theo p. 9/ integrate from to α> T (α) T (α) = 2α + i 2 U(α) ln T (α) = ln(α /2 ) + i 2 α U(α)dα

30 Theo p. 9/ integrate from to α> T (α) T (α) = 2α + i 2 U(α) ln T (α) = ln(α /2 ) + i 2 polar representation of T (α) T (α) = ( i α exp 2 α α U(α)dα ) U(α)dα, α >

31 Theo p. 9/ integrate from to α> T (α) T (α) = 2α + i 2 U(α) ln T (α) = ln(α /2 ) + i 2 polar representation of T (α) T (α) = ( i α exp 2 α α U(α)dα ) U(α)dα, α > since T () = + i U(), the slope of the tangent vector to the 2 2 spiral at α = is U() = (Theodorus constant) k 3/2 + k /2

32 Theo p. 9/ integrate from to α> T (α) T (α) = 2α + i 2 U(α) ln T (α) = ln(α /2 ) + i 2 polar representation of T (α) T (α) = ( i α exp 2 α α U(α)dα ) U(α)dα, α > since T () = + i U(), the slope of the tangent vector to the 2 2 spiral at α = is U() = (Theodorus constant) k 3/2 + k /2 numerical analysis and special functions: compute and identify U(α) = U(α)dα for < α < 2, (k+α ) 3/2 +(k+α ) /2 α

33 Theo p. / first digression summation by integration (G. & Milovanović, 985) s = a k, a k = (Lf)(k)

34 Theo p. / first digression summation by integration (G. & Milovanović, 985) s = a k, a k = (Lf)(k) s = (Lf)(k) = e kt f(t)dt

35 Theo p. / first digression summation by integration (G. & Milovanović, 985) s = a k, a k = (Lf)(k) = s = (Lf)(k) = ( ) e kt f(t)dt = e kt f(t)dt t e t f(t) t dt

36 Theo p. / first digression summation by integration (G. & Milovanović, 985) s = a k, a k = (Lf)(k) Thus = s = (Lf)(k) = ( ) e kt f(t)dt = a k = ε(t) = t e t f(t) t ε(t)dt, e kt f(t)dt t e t f(t) t f = L a Bose Einstein distribution dt

37 Theo p. / Theodorus: a k = k /2 = k 3/2 + k/2 k + convolution theorem for Laplace transform Lg Lh = Lg h, (g h)(t) = t g(τ)h(t τ)dτ

38 Theo p. / Theodorus: a k = k /2 = k 3/2 + k/2 k + convolution theorem for Laplace transform Lg Lh = Lg h, (g h)(t) = t g(τ)h(t τ)dτ application to a k k /2 = k + ( L t /2 π ) (k) = ( L e t) (k)

39 Theo p. / Theodorus: a k = k /2 = k 3/2 + k/2 k + convolution theorem for Laplace transform Lg Lh = Lg h, (g h)(t) = t g(τ)h(t τ)dτ application to a k k /2 = k + ( L t /2 π ) (k) = ( L e t) (k) a k = ( L t /2 π ) (k) (L e t ) (k) = t (L π ) τ /2 e (t τ) dτ (k)

40 Theo p. 2/ Theodorus (cont ) f(t) = π e t t = 2 π e t t τ /2 e τ dτ e x2 dx = 2 π F ( t)

41 Theo p. 2/ Theodorus (cont ) f(t) = π e t t = 2 π e t t τ /2 e τ dτ e x2 dx = 2 π F ( t) k 3/2 + k /2 = f(t) t ε(t)dt = 2 π F ( t) t w(t)dt

42 Theo p. 2/ Theodorus (cont ) f(t) = π e t t = 2 π e t t τ /2 e τ dτ e x2 dx = 2 π F ( t) k 3/2 + k /2 = f(t) t ε(t)dt = 2 π F ( t) t w(t)dt w(t) = t /2 ε(t) = t/2 e t

43 Theo p. 3/ second digression Gaussian quadrature n-point quadrature formula g(t)w(t)dt = n λ (n) k g(τ (n) k ), g P 2n

44 Theo p. 3/ second digression Gaussian quadrature n-point quadrature formula g(t)w(t)dt = n λ (n) k g(τ (n) k ), g P 2n orthogonal polynomials (π k, π l ) =, k l, where (u, v) = u(t)v(t)w(t)dt

45 Theo p. 3/ second digression Gaussian quadrature n-point quadrature formula g(t)w(t)dt = n λ (n) k g(τ (n) k ), g P 2n orthogonal polynomials (π k, π l ) =, k l, where (u, v) = three-term recurrence relation u(t)v(t)w(t)dt π k+ (t) = (t α k )π k (t) β k π k (t), k =,, 2,... π (t) =, π (t) = where α k = α k (w) R, β k = β k (w) >, β = w(t)dt

46 Theo p. 4/ Jacobi matrix J n (w) = α β β α β 2 β 2 α β n β n α n

47 Theo p. 4/ Jacobi matrix J n (w) = α β β α β 2 β 2 α β n β n α n Gaussian nodes and weights (Golub & Welsch, 969) τ (n) k = eigenvalues of J n, λ (n) k = β v 2 k, v k, = first component of (normalized) eigenvector v k

48 Theo p. 4/ Jacobi matrix J n (w) = α β β α β 2 β 2 α β n β n α n Gaussian nodes and weights (Golub & Welsch, 969) τ (n) k = eigenvalues of J n, λ (n) k = β v 2 k, v k, = first component of (normalized) eigenvector v k moments of w µ k = t k w(t)dt, k =,,..., 2n

49 Theo p. 4/ Jacobi matrix J n (w) = α β β α β 2 β 2 α β n β n α n Gaussian nodes and weights (Golub & Welsch, 969) τ (n) k = eigenvalues of J n, λ (n) k = β v 2 k, v k, = first component of (normalized) eigenvector v k moments of w µ k = Chebyshev algorithm t k w(t)dt, k =,,..., 2n {µ k } 2n k= {α k, β k } n k=

50 Theo p. 5/ numerical results for Gaussian quadrature (in 5D-arithmetic) µ k = t k w(t)dt = t k+/2 e t dt = Γ(k + 3/2)ζ(k + 3/2)

51 Theo p. 5/ numerical results for Gaussian quadrature (in 5D-arithmetic) µ k = t k w(t)dt = t k+/2 e t dt = Γ(k + 3/2)ζ(k + 3/2) k 3/2 + k = 2 [F ( t)/ t ] w(t) /2 π s n = 2 n ( ) λ (n) π k F τ (n) k / τ (n) k

52 Theo p. 5/ numerical results for Gaussian quadrature (in 5D-arithmetic) µ k = t k w(t)dt = t k+/2 e t dt = Γ(k + 3/2)ζ(k + 3/2) k 3/2 + k = 2 [F ( t)/ t ] w(t) /2 π s n = 2 n ( ) λ (n) π k F τ (n) k / τ (n) k n s n

53 Theo p. 6/ computation and identification of U(α) and α U(α)dα (k + α ) /2 (k + α ) + = t ) (L τ /2 e (t τ) dτ (k + α ) π

54 Theo p. 6/ computation and identification of U(α) and α U(α)dα (k + α ) /2 (k + α ) + = t ) (L τ /2 e (t τ) dτ (k + α ) π applying the shift property for Laplace transform yields (Lg) (s + b) = ( L e bt g(t) ) (s) (k + α ) /2 (k + α ) + = t L (e αt π ) τ /2 e τ dτ (k)

55 Theo p. 6/ computation and identification of U(α) and α U(α)dα (k + α ) /2 (k + α ) + = t ) (L τ /2 e (t τ) dτ (k + α ) π applying the shift property for Laplace transform yields hence (Lg) (s + b) = ( L e bt g(t) ) (s) (k + α ) /2 (k + α ) + = t L (e αt π f(t) = π e αt t ) τ /2 e τ dτ (k) τ /2 e τ dτ = 2 π e (α )t F ( t)

56 Theo p. 7/ U(α) = f(t) t ε(t)dt = 2 e (α )t F ( t) π t w(t)dt w(t) = t/2 e t, α < 2

57 Theo p. 7/ U(α) = f(t) t ε(t)dt = 2 e (α )t F ( t) π t w(t)dt w(t) = t/2 e t, α < 2 identification of U(α) as a Laplace transform U(α) = (Lu) (α ), u(t) = 2 π F ( t) t w(t)

58 Theo p. 7/ U(α) = f(t) t ε(t)dt = 2 e (α )t F ( t) π t w(t)dt w(t) = t/2 e t, α < 2 identification of U(α) as a Laplace transform U(α) = (Lu) (α ), u(t) = 2 π F ( t) t w(t) the integral of U(α) α U(α)dα = 2(α ) π e (α )t (α )t F ( t) t w(t)dt

59 Theo p. 8/ Epilogue summation theory can be generalized to series s + = k ν r(k), s = ( ) k k ν r(k), < ν < where r(k) is a rational function

60 Theo p. 8/ Epilogue summation theory can be generalized to series s + = k ν r(k), s = ( ) k k ν r(k), < ν < where r(k) is a rational function regularizing transformation (J. Waldvogel, in preparation) α = r 2, r R

61 Theo p. 8/ Epilogue summation theory can be generalized to series s + = k ν r(k), s = ( ) k k ν r(k), < ν < where r(k) is a rational function regularizing transformation (J. Waldvogel, in preparation) then T (α) = T (r 2 ) = α = r 2, + i + i/r k=2 r R + i/ k + i/ r 2 + k is analytic for r C \ [i, i ] [ i, i ]

62 Theo p. 8/ Epilogue summation theory can be generalized to series s + = k ν r(k), s = ( ) k k ν r(k), < ν < where r(k) is a rational function regularizing transformation (J. Waldvogel, in preparation) then T (α) = T (r 2 ) = α = r 2, + i + i/r k=2 r R + i/ k + i/ r 2 + k is analytic for r C \ [i, i ] [ i, i ] r > : spiral of Theodorus

63 Theo p. 8/ Epilogue summation theory can be generalized to series s + = k ν r(k), s = ( ) k k ν r(k), < ν < where r(k) is a rational function regularizing transformation (J. Waldvogel, in preparation) then T (α) = T (r 2 ) = α = r 2, + i + i/r k=2 r R + i/ k + i/ r 2 + k is analytic for r C \ [i, i ] [ i, i ] r > : spiral of Theodorus r < : analytic continuation of the spiral

64 Theo p. 9/ twin-spiral of Theodorus

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