MATHEMATICAL PUBLICATIONS

August 22, 2015

Thomas J Osler

0. Osler’s Doctoral Thesis: Leibniz rule, the chain rule and Taylor’s theorem for fractional derivatives, February, 1970, The Courant Institute, New York University. Advisor: Samuel Karp. Download PDF file

1. Leibniz rule for fractional derivatives generalized and an application to infinite

2. The fractional derivative of a composite function, SIAM J. Math. Anal, 1 (1970),

3. Taylor’s series generalized for fractional derivatives and applications, SIAM J.

4. Fractional derivatives and leibniz rule, Amer. Math. Monthly, 78 (1971),

5. A further extension of Leibniz rule to fractional derivatives and its relation to Parseval’s formula, SIAM J. Math. Anal., 3 (1972), pp. 1-16. MR 48#2323a download PDF file

6. An integral analogue of Taylor’s series and its use in computing Fourier
transforms, Math. Comp., 26 (1972), pp. 449-460. MR 46#5950. download PDF file

7. The integral analog of the Leibniz rule, Math. Comp., 26 (1972), pp. 903-915.

8. A correction to Leibniz rule for fractional derivatives, SIAM J. Math. Anal., 4 (1973), pp. 456-459. MR 48#2323b download PDF file

9. Differences of arbitrary order (With J. B. Diaz), Math. Comp., 28 (1974),

10. Fundamental properties of fractional derivatives via Pochhammer integrals.(With J. L. Lavoie and R. Tremblay). A paper in "Fractional Calculus and Its Applications" edited by B. Ross, Springer-Verlag, New York, 1975, pp. 323-356.
MR 57#16490

11. Open questions for research. (Proc. Internat. Conf. Univ. New Haven, West
Haven
, Conn.
, 1974) pp. 376-381. Lecture Notes in Math., Vol. 457, Springer,
Berlin, 1978. (A. M. Brickner) MR 57#16493

12. An indentity for simplifying certain generalized hypergeometric functions, Math Comp., 29 (1975), pp. 888-893. MR 51#10710 download PDF file

13. Leibniz rule for fractional derivatives used to generalize formulas of Walker and Cauchy, Bul. Inst. Polit, Jasi, 21-(25), 1-2, (1975), pp. 21-24. MR 52#8822 download PDF file

14. Fractional derivatives and special functions, SIAM Review, 18 (1976),

15. A quick look at Lyapunov space, Mathematics and Computer Education,

16. A mathematical art exhibit, Mathematics and Computer Education, 29 (1995),
pp. 245-252.

17. Get billions and billions of digits of pi from a wrong formula, Mathematics and Computer Education , Vol. 33, No. 1, 1999, pp.40-45. download PDF version

20. The tautochrone under arbitrary potentials using fractional derivatives, (with Eduardo Flores), American Journal of Physics, (Am. J. Phys.), 67(1999), pp. 718- 722. download PDF version

23. A child’s garden of fractional derivatives, (with Marcia Kleinz), The College Mathematics Journal, Vol. 31, No. 2, (2000), pp. 82-887 download PDF version

25. Spacetime numbers the easy way, (with Nick Borota and Eduardo Flores), Mathematics and Computer Education, Vol. 34, No. 2, (2000), pp. 159-168. download PDF version

26. Playing with partitions on the computer, (with Abdul Hassen), Mathematics and Computer Education , Vol. 35, No. 1, (2001), pp. 5-17. download PDF version

27. Mutations of the Mandelbrot set, (with Jae Hattrick-Simpers, student), Mathematics and Computer Education, Vol. 35, No. 1, (2001), pp. 18-26. download PDF version

28. The remarkable incircle of a triangle. (with Ira Fine), Mathematics and Computer Education, Vol. 35, No. 1, (2001), pp. 44-50. download PDF version

30. The rotating tautochrone, (with Eduardo Flores), Journal of Applied Mechanics,

36.   A tale of two series, (with Marcus Wright), The College Mathematics Journal, 33(2002), pp. 99-106. download PDF file

37.  Divergence of the harmonic series, p series and others by rearrangement, (with Bernard August, Emeritus), The College Mathematics Journal, 33(2002), pp. 233-4. download PDF version

39. A table of the partition function, (with Abdul Hassen and T. R. Chandrupatla), The Mathematical Spectrum, 34(2001/2002), pp. 55-57. download PDF version

40. Visual Basic for QuickBasic Programmers, (with Seth Bergmann and T. R. Chandrupatla), Mathematics and Computer Education, 36(2002), pp. 133-138.  download PDF file

41. Functions of a spacetime variable, (with Nicolae A. Borota), Mathematics and Computer Education, 36(2002), pp. 231-239. Download PDF file

42. An easy introduction to biplex numbers, (with Dipti Bardhan), Mathematics and Computer Education, 36(2002), pp. 278-286. Download PDF file

43. An easy look at the cubic formula, Mathematics and Computer Education, 36(2002), pp. 287-290. Download PDF file

44. Proof without words: Integral of sine squared, The AMATYC Review, 24(2002), p. 65.  Download PDF file

45.  A computer hunt for Apery's constant, (with Brian Seaman), Mathematical Spectrum, 35(2002/2003), No.1, pp. 5-8.  Download PDF file

46. A magic trick from Fibonacci, (with James Smoak), The College Mathematics Journal, 34(2003), pp. 58-60.  Download PDF file

47.  Variations on a theme from Pascal's triangle,  The College Mathematics

48.  Visual proofs of two integrals, The College Mathematics Journal, 34(2003), p.231-232.. Download PDF file

49. An unusual product for sin z and variations of Wallis’s product, The Mathematical Gazette, 87(2003), pp. 134-139.  Download PDF file

50. Finding Zeta(2p) from a product of sines. The American Mathematical Monthly, 111(2004), pp. 52-54.  Download (PDF) original version containing motivational material.

51. Another intuitive approach to Stirling’s formula, International Journal of Mathematical Education in Science and Technology, 35(2004), pp.111-118. Download PDF file.

52. Geometric construction of Pythagorean triangles, (with Tirupathi R. Chandrupatla), Mathematics and Computer Education, 38(2004), pp. 90-91.

53. Variations on the linear first order ODE, (with Brian Seaman), International Journal of Mathematical Education in Science and Technology, 35(2004), pp.309-315. Download (PDF) version.

54. Extending Theon's ladder to any square root, (with Shaun Giberson), The College Mathematics Journal, 35(2004), pp. 222-226. Download PDF file

55. Problems on divisibility of binomial coefficients, (with James Smoak). The AMATYC Review, 25(2004), pp. 17-21.Download PDF file.

56. Generalizing integrals involving x^x and series involving n^n, (with Jeffrey Tsay). Mathematics and Computer Education, 39(2005), pp. 31-36. Download PDF File

57. Theon's ladder for any root (with Marcus Wright and Michael Orchard, International Journal of Mathematical Education in Science and Technology, 36(2005), pp. 389-398. Download PDF file

58. Why sin 80x looks like sin x on my TI-89, (with Jim Zeng).International Journal of Mathematical Education in Science and Technology, 36(2005), pp. 437-441. Download PDF file

59. Unexpected constructable numbers, The AMATYC Review, 26(2005), no. 2, pp. 2-3. Download PDF file

60. Some unusual expressions for the inradius of a triangle, (with Tirupathi R. Chandrupatla), The AMATYC Review, 26(2005), no.2, pp. 12-17. Download PDF file

61.  A simple geometric construction of the harmonic mean of n variables (with Jim Zeng),  Mathematical Spectrum, 37(2004/2005), No. 3, pp. 109-111. Download PDF file

62. Proof Without Words: Arc length of the cycloid,  The Mathematical Gazette, 89(2005), no. 515, page 250. Download PDF file.

63. Ptolmey vs. Copernicus. (With Joseph Diaco.) The  Mathematical Spectrum, 38(2005/2006), pp. 3-6.  Download PDF File

64.  The general Vieta-Wallis product for pi, The Mathematical Gazette, 89(2005), pp. 371-378. Download PDF File

65. Using simple harmonic motion to help in the search for tautochrone curves, (with Tirupathi R. Chandrupatia), International Journal of Mathematical Education in Science and Technology, 37 ( 2006),  pp, 104-9. Download PDF file

66.  Motivation for finding Zeta(2p) from a product of sines,  International Journal of Mathematical Education in Science and Technology  37(2006), pp. 231-235.  Download PDF file

67. A proof of the continued fraction for e^1/M, American Mathematical Monthly, 113(2006), pp. 62-66. Download PDF file

68. A mechanical view of Archimedes quadrature of the parabola. The College Mathematics Journal, 37(2006), pp. 24-28. Download PDF file

69.  Interesting finite and infinite products from simple algebraic identities. The Mathematical Gazette, 90(2006), pp. 90-93. Download PDF File.

70  Proof with few words: Quadratic convergence of the AGM, to appear in The Mathematical Gazette 90(2006), pp. 116-117. Download PDF File.

71. A spiral of triangles related to the great pyramid, with Matthew Oster. The Mathematical Spectrum 38(2005/2006), pp. 108-112,. Download PDF File

72. An asymptotic approach to constructing the hyperbola, with David Grochowski. The Mathematical Spectrum 38(2005/2006), pp. 113-115.  Download PDF File

73.. A collection of numbers whose proof of irrationality is like that of the number e, (with Nicholas Stugard). Mathematics and Computer Eduation, 40(2006), pp. 103-107. Download PDF file

74. Translation with Notes of Euler's  paper "Remarks on a beautiful relation between direct as well as reciprocal series" (E352),(with Lucas Willis), on the web at the Euler Archive, http://www.math.dartmouth.edu/~euler/.  Download PDF FIle

75. Synopsis of Euler's paper "Remarks on a beautiful relation between direct as well as reciprocal series" (E352),(with Lucas Willis), on the web at the Euler Archive http://www.math.dartmouth.edu/~euler/.  Download PDF file

76. Translation with notes of Euler’s paper E796 ( with Kristen McKeen), Recherches sur le problem de trois nombres carres tels que la somme de deux quelconques moins le troisieme fasse un nombre carre,“Research into the problem of three square numbers such that the sum of any two less the third one provides a square number.”, on the web at the Euler Archive http://www.math.dartmouth.edu/~euler/. Download PDF file  .

77. Synopsis by section of Euler's paper E769 ( with Kristen McKeen), Recherches sur le problem de trois nombres carres tels que la somme de deux quelconques moins le troisieme fasse un nombre carre,“Research into the problem of three square numbers such that the sum of any two less the third one provides a square number”, on the web at the Euler Archive http://www.math.dartmouth.edu/~euler/.   Download PDF file

78. Translation with notes of Euler's paper E46, Methodus Universalis Serierum Convergentium Summas Quam Proxime Inveniendi, (A general method for finding approximations to the sums of convergent series). (with Walter Jacob.) On the web at the Euler Archive http://www.math.dartmouth.edu/~euler/.  Download PDF file

79. Synopsis of Euler's paper E46, Methodus Universalis Serierum Convergentium Summas Quam Proxime Inveniendi, (A general method for finding approximations to the sums of convergent series). (with Walter Jacob.) on the web at the Euler Archive http://www.math.dartmouth.edu/~euler/. Download PDF file

80.  The integers of James Booth, (with John F. Kennedy), Mathematical Spectrum, 39(2006/2007), pp. 71-72. Download PDF file

81. A simple geometric method of estimating the error in using Vieta's product for pi, International Journal of Mathematics Education in Science and Technology, (38)2007, pp. 136-142. Download PDF file

82. Some cosine relations and the regular heptagon, ( with Phongthong Heng) Mathematics and Computer Education, (41)2007, pp. 17-21. Download PDF file

83. Developing formulas by skipping rows in Pascal's triangle (with Robert J. Buonpastore). Mathematics and Computer Education, (41)2007, pp. 25-29.  Download PDF file

84. Some long telescoping series. The Mathematical Gazette, 91(2007), pp. 104-5. Download PDF file

85.  Finding Zeta(2n) from a recursion relation for Bernoulli numbers (with Jim Zeng), The Mathematical Gazette, 91(2007), pp. 123-6. Download PDF file

86. The quadratic equation as solved by Persian mathematicians of the middle ages, (with Mohamed Teymour). Mathematical Spectrum, 39(2006/2007), pp. 115-8. Download PDF file

87. Lucky fractions: Where bad arithmetic gives correct results, Mathematics and Computer Education, 41(2007), pp.162-167.Download PDF file

88. Constructing a line segment whose length is equal to the measure of a given angle, (with Walter Jacob, IV), International Journal of Mathemeatical Education in Science and Technology, 38(2007), pp. 529-230. Download PDF file

89.  The unlikely connection between partitons and divisors, (with Abdul Hassen and T. R. Chandrupatla), The College Mathematics Journal, 38(2007), pp. 278-287. Download PDF FIle

90. Oblique-angled diameters and the conic sections, with Edward Greve. The Mathematical Spectrum, 40(2007/2008), pp. 26-30.. Download PDF File

91. Geometric constructions approximating pi. Mathematical Spectrum, 40(2007/2008), pp106-108. Download PDF File

92. Vieta-like products of nested radicals with Fibonacci and Lucas numbers,  Fibonacci Quarterly, 45(2008), pp. 202-204.  Download PDF File

93. An unusual proof that Fm divides Fmn for Fibonacci numbers using hyperbolic functions, (with Adam Hilburn). The Mathematical Gazette, 91(2007), pp. 510-512.  Download PDF File

94. Another geometric vision of the hyperbola. Mathematical Spectrum  41(2008/2009). Pp. 123-4.  . Download PDF File

95.  Approximating the path of a celestial body with a circular orbit from two close observations, (with Joseph Palma), The Mathematical Scientist, 33(2008), pp. 8-14.  Download PDF File

96. Euler's little summation formula and sums of powers, (with Andrew Robertson). Mathematical Spectrum, 40(2006/2007), pp. 73-76.  Download PDF File

97. Euler's little summation formula and special values of the zeta function, The Mathematical Gazette, 92(2008), pp. 295-299. Download PDF File

98. Exact values of the hyperbolic functions, Mathematics and Computer Education, 42(2008), pp. 58-59.  Download PDF File

100. Translation with notes of Euler's paper E83, (with Edward Greve)On some properties of   conic sections that are shared with infinitely many other curved lines.  on the web at the Euler Archive http://www.math.dartmouth.edu/~euler/. Download PDF File

101. Synopsis of Euler's paper E83, On some properties of  conic sections that are shared with infinitely many other curved lines. on the web at the Euler Archive http://www.math.dartmouth.edu/~euler/. Download PDF File

102. Triangles and parallelograms of equal area inside the ellipse, (with Robert Buonpastore).  Mathematical Spectrum. 41(2008/2009), pp. 23-26. Download PDF FIle

103. On generalizations of Lambert’s series, (with Abdul Hassen), International Journal of Pure and Applied Mathematics, 43(2008), 465-484.  Download PDF FIle

104. Visualizing Vandermonde's convolution on the extended Pascal's triangle, (with Tirupathi R. Chandrupatia). Mathematics and Computer Education, 42(2008), pp. 118-122. Download PDF File

105. A remarkable formula for approximating the sum of alternating series. The Mathematical Gazette, 93(2009), pp.76-82. Download PDF FIle

106. Approximating an ellipse with four circular arcs, (with Tirupathi R. Chandrupatia), Mathematics and Computer Education, 42(2008), pp. 220-228. Download PDF File

107. More exact values of the hyperbolic functions, Mathematics and Computer Education, 42(2008), pp. 193-197. Download PDF File

110. On numbers equal to the sum of two squares in more than one way,  (with Ming-Sun Li, Kathryn Robertson, Abdul Hassen, Christopher S. Simons and Marcus Wright). Mathematics and Computer Education, 43(2009), pp. 102-108. Download PDF File

111. Conjectures from a historic table by John Wallis, with Lee Collins. Mathematical Spectrum, 42(2009/2010), pp. 14-19. Download PDF File

113. Euler and the functional equation for the zeta function.The Mathematical Scientist. 34( 2009), pp. 62-73. Download PDF File

114 The tables of John Wallis and the discovery of his product for pi. The Mathematical Gazette, 94(2010), pp. 430-437. Download PDF file

115. Five historic formulas for pi. Mathematics and Computer Education, 43(2009), pp. 250-258. Download PDF File

117. Lord Brouncker’s forgotten sequence of continued fractions for pi. International Journal of Mathematical Education in Science and Technology, Volume 41, Issue 1 January 2010 , pages 105 – 110. Download PDF File

118. Translation with notes of Euler's paper E745, (with Christopher Tippie and Kristin Masters) On the continued fractions of Wallis.  on the web at the Euler Archive http://www.math.dartmouth.edu/~euler/. Download PDF FIle

119. Reflections and Notes on Euler’s paper E745, (with Christopher Tippie and Kristin Masters) On the continued fractions of Wallis.  on the web at the Euler Archive http://www.math.dartmouth.edu/~euler/. Download PDF File

120. Synopsis of Euler’s paper E745, (with Christopher Tippie and Kristin Masters) On the continued fractions of Wallis.  on the web at the Euler Archive http://www.math.dartmouth.edu/~euler/. Download PDF File

121. .  Euler’s parallel oblique angled diameters,  The Mathematical Gazette, 95(2011), pp. 40-48. Download PDF file

122. Translation of Euler's paper , (with Jasen Scaramazza) Memoire sur la plus grande equation des planets, (Memoir on the Maximum value of an Equation of the Planets). on the web at the Euler Archive http://www.math.dartmouth.edu/~euler/. Download PDF file

123. Notes on  Euler's paper , (with Jasen Scaramazza) Memoire sur la plus grande equation des planets, (Memoir on the Maximum value of an Equation of the Planets). on the web at the Euler Archive http://www.math.dartmouth.edu/~euler/. Download PDF file

124. Synopsis of Euler's paper , (with Jasen Scaramazza) Memoire sur la plus grande equation des planets, (Memoir on the Maximum value of an Equation of the Planets). on the web at the Euler Archive http://www.math.dartmouth.edu/~euler/.  Download PDF file

125 The perimeter of an ellipse, (with Tirupathi R. Chandrupatla).The Mathematical Scientist, 35 (2010), pp. 122-131. Download PDF file

126. An interesting generalization of the half angle formulas to complex numbers, (with Marcus Wright), The Mathematical Gazette, 95(2011), p. 327.  Download PDF file

127. Euler’s method of integration by parts   (with Steve Donahue), The Mathematical Gazette.  96(2012), pp. 115-116. Download PDF file

128. Planar biarc curves – A geometric view, (with T. R. Chandrupatla). The Mathematical Scientist. 36(2011), pp. 126-132.  Download PDF file

129. Law of cosines generalized for any polygon and any polyhedron, (with Lee Collins), The Mathematical Gazette, 95(2011), pp. 240-243. Download PDF file

130. The missing fractions in Brouncker’s sequence of continued fractions for pi. The Mathematical Gazette, 96(2012), pp. 221-225. Download PDF file

131. How Euler found the eccentricity of planetary orbits, (with Jasen Scaramazza). The Mathematical Scientist, 37(2012), pp. 89-95.. Download PDF file

132. Diophantine approximation and the irrationality of certain numbers, (with Eric Jones). Mathematics and Computer Education, 46(2012), pp. 125 - 130. Download PDF file

133. Partial sums of divergent series that cannot be an integer. The Mathematical Gazette, 96(2012), pp. 515-519. .Download PDF file

134. The devil’s series – Did it fool Euler. (with Steven Donahue), The Mathematical Gazette, 97(2013), pp. 133 – 137. Download PDF file

135. An interesting bilateral series generalizing the series for zeta of 2, (with Cory Wright), The Mathematical Gazette, 97(2013), pp. 131 – 133.  Download PDF fi;e

136. Unexpected connections between three famous formulas for pi. Submitted via email to the Mathematical Scientist on March 25, 2012. Accepted on April 13,  Download PDF file

137. Four derivations of an interesting bilateral series generalizing the series for zeta of 2. (with Cory Wright). The International Journal of Mathematical Education in Science and Technology, 44(2013), pp. 456 – 461. Download PDF file

138. Interesting subseries of the harmonic series.  Mathematics and Computer Education, 46(2012), pp. 169 – 173. Download PDF file

139. Recursive formulas related to the AGM, (with Tirupathi R. Chandrupatla). The Mathematical Gazette, 98(2014), pp. 484-486.. Download PDF file

140  Vieta’s product for Pi from Archimedes, The Mathematical Gazette, 98(2014) pp. 429-431. Download PDF file

141  Can divergent series be of value?. The Mathematical Scientist, 39(2014), pp. 100-106. Download PDF file

142  A higher look at vertical motion The Mathematical Scientist, 39(2014), pp. 138-142. Download PDF file

143  A novel approach to integral csc x, (with Walter Jacob). The Mathematical Gazette, 98(2014), pp. 518-519. Download PDF file

144  A product of nested radicals for the AGM., Accepted July 20, 2014 by The American Mathematical Monthly. Due to appear in mid to late 2015, Download PDF file

145 Short and fuzzy derivations of five remarkable formulas for primes, The Mathematical Gazette, 99(2015), pp. 220 - 225.

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146 Iterations for the lemniscate constant resembling the Archimedean algorithm for pi. Accepted by The American Mathematical Monthly on  May 27, 2015. Submitted on January 4, 2015.

147.   A table of definite integrals from the marriage of power and Fourier series, (with James Rosado). Accepted on June 24, 2015 by  Revista Scientia.  .