__MATHEMATICAL
PUBLICATIONS __

April 2, 2014

Thomas J Osler

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

*series*, SIAM J. Appl. Math., 18 (1970), pp. 658-674. MR 41#5562
(download from JSTOR) download PDF file

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

pp. 288-293. MR 41#5563 download
PDF file

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

Math. Anal. 2 (1971), pp. 37-48. MR 45#3682 download PDF file

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

pp. 645-649. (download from JSTOR) download
PDF file

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 *

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

MR 47#2792. download PDF file

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),

pp. 185-202. MR 49#11077 (download from JSTOR) download PDF file

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.

Haven

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),

pp. 240-268. MR 53#13666 download PDF
file

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

28 (1994), pp. 183-197.

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

18. *The trinomial triangle, *(with James Chappell, student), The
College Mathematics Journal, Vol. 30, No. 2, 1999, pp. 141-142. download PDF version

19. *Extending the Babylonian algorithm, *Mathematics and Computer
Education, Vol. 33, No. 2, 1999, pp. 120-128. 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

21. *The united Vieta’s and Wallis’s products
for pi,* American Mathematical Monthly, 106 (1999), pp. 774-776. download PDF version

22. *Fractal Movies,* Mathematics and Computer Education, Vol. 33
(1999), pp. 236-243. 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

*24. The binomial to binomial theorem,* The College Mathematics
Journal, Vol. 31, No. 3, (2000), pp. 211-212. 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

*29. Cardan polynomials and the reduction of
radicals, *Mathematics Magazine, Vol 47, No. 1,
(2001), pp. 26-32. download PDF version

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

68(2001), pp. 353-356. download PDF version

31. *Rearranging terms of a harmonic-like series, *Mathematics and
Computer Education, 35(2001), pp. 136-139. download
PDF version

32. *Fun with 0.999...* , The AMATYC Review, 22(2001), pp. 53-56. download PDF version

33. A*n unusual approach to Kepler’s first
law*, American Journal of Physics, 69(2001), pp. 1036-8. download PDF version

34. *An unusual view of the ellipse*, Mathematics and Computer
Education, 35,(2001), pp. 234-236. download PDF version

35. *Variations on Vieta’s and Wallis’s products
for pi,* (with Michael Wilhelm), Mathematics and Computer Education,
35(2001), pp. 225-232. download PDF version

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

38. *Fermat’s little theorem from the multinomial theorem*, The College
Mathematics Journal, 33(2002), p. 239. 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

Journal, 34(2003), pp. 216-223. Download PDF file

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.

Download (PDF) final accepted version (more
formal with less motivation)

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.

Download PDF file

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.

90.

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

99. *Using Theon's ladder to find roots of
quadratic equations*, Mathematics and Computer Education, 42(2008),
pp. 23-27. 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

*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

108. *A modern look at a neglected summation
formula by Euler*, (with Walter Jacob). The Mathematical Gazette, 93(2009),
pp. 237-243. Download
PDF file

109. *Vieta**-like products of nested radicals*, The
Mathematical Gazette, 94(2010), pp. 62-66. 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

112. *Triangles of equal area in a hyperbola*, with Adam Romasko,
Mathematical Spectrum, 42(2009/2010), pp. 70-74. 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

116. * Morphing Lord Brouncker’s
Continued Fraction for pi into the Product of Wallis*. The Mathematical Gazette 95(2011), pp. 17-22.

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. N*otes 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.*
Accepted on January 24, 2012. Due to appear November 2012 in the Mathematical
Gazette. 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 *Vieta’s** product for
Pi from Archimedes, *Submitted to The
Mathematical Gazette on January 6, 2014. Revision requested on February 14,
2014 and revised paper sent on March 23, 2014. Accepted on March 24, 2014. Due
to appear November 2014. Download
PDF file

140 *Can
divergent series be of value?. *Submitted to The Mathematical Scientist on
March 26, 2014 via email. Accepted April 2, 2014 Download
PDF file