Technical Reference Mathematics Complex Formulae / Roots and Powers Chart / Important Mathematical Constants / Polyhedra / Spherical Harmonics /

Complex Formulae

1) z = x + iy where x = Real part of z and y = Imaginary part of z

2) c = a + ib where a = Real part of c and b = Imaginary part of c

3) z = re^iq = (sqrt(x^2 + y^2)) (cos q + i sin q)
where q = arctan (y / x), r = sqrt(x^2 + y^2) and "sqrt" means square root

4) z^n = r^n*e^inq = (sqrt(x^2 + y^2))^n (cos nq + i sin nq) ; r and q as above

5) sqrt(z) = (sqrt(r)sqrt(e^iq)) = (sqrt(sqrt(x^2 + y^2))) [cos (.5 arctan (y / x))
+ i sin (arctan (y / x))]

6) ln z = ln[sqrt(x^2 + y^2)] + i arctan (y / x)

7) e^z = e^x(cos y + i sin y)

8) sin z = sin x cosh y + i cos x sinh y = -i sinh iz = (e^iz - e^-iz) / 2i

9) cos z = cos x cosh y - i sin x sinh y = cosh iz = (e^iz + e^-iz) / 2

10) sinh z = - i sinh iz = (e^z - e^-z) / 2

11) cosh z = cos iz = (e^z + e^-z) / 2

12) tanh z = - i tan (iz) = (e^z - e^-z) / (e^z + e^-z)

13) sech z = sech (iz) = [cosh z] ^ -1

14) csch z = i csc (iz) = [sinh z] ^ -1

15) arcsinh z = ln(z + sqrt(z^2 + 1))

16) arccosh z = ln(z + sqrt(z^2 - 1)) , ln(z - sqrt(z^2 - 1))

17) arctanh z = .5 * ln[(1 + z) / (1 - z)]

18) arcsech z = ln[(1 + sqrt(z^2 + 1)) / z]

19) arccsch z = ln[(1 + sqrt(1 - z^2 )) / z] , ln[(1 - sqrt(1 - z^2 )) / z]

20) arccoth z = .5 * ln[(z + 1) / (z - 1)]

21) sin^2(z) + cos^2(z) = 1

22) cosh^2(z) - sinh^2(z) = 1

23) tan z = (sin 2x + i sinh 2y) / (cos 2x + cosh 2y)

24) cot z = (sin 2x - i sinh 2y) / (cosh 2y - cos 2x)

25) nth root of z = [nth root of (x^2 + y^2)](cos (q / n) + i sin (q / n))

26) Newton's Method z(n+1) = z(n) - [f(z(n)) / f '(z(n))]

27) Henon Attractor: (for z(n) = x(n) + iy(n)) , x(n+1) = ax(n) + y(n) and y(n+1)= bx(n)

28) Halley Map: z(n+1) = z(n) - L[(2f(z(n))f '(z(n))) / (2(f '(z(n)))^2 - f' '(z(n))f(z(n)))]

29) Lorenz Attractor: dx / dt = a(y - x) dy / dt = x(r - z) - y dz / dt = xy - bz

Roots and Powers Chart

 N N^2 N^3 sqrt(N) N N^2 N^3 sqrt(N) 1 1 1 1 21 441 9261 4.583 2 4 8 1.414 22 484 10648 4.690 3 9 27 1.732 23 529 12167 4.796 4 16 64 2 24 576 13824 4.899 5 25 125 2.236 25 625 15625 5 6 36 216 2.449 26 676 17576 5.099 7 49 343 2.646 27 729 19683 5.196 8 64 512 2.828 28 784 21952 5.292 9 81 729 3 29 841 24389 5.385 10 100 1000 3.162 30 900 27000 5.477 11 121 1331 3.317 31 961 29791 5.568 12 144 1728 3.464 32 1024 32768 5.657 13 169 2197 3.606 33 1089 35937 5.745 14 196 2744 3.742 34 1156 39304 5.831 15 225 3375 3.873 35 1225 42875 5.916 16 256 4096 4 36 1296 46656 6 17 289 4913 4.123 37 1369 50653 6.083 18 324 5832 4.243 38 1444 54872 6.164 19 361 6859 4.359 39 1521 59319 6.245 20 400 8000 4.472 40 1600 64000 6.325

Important Mathematical Constants

1) Pi --- The ratio of the circumference of a circle to its diameter, supposedly first discovered by Archimedes (287-212 BC). He surmised that pi was

3 10/17 < Pi < 3 1/7

The first hundred digits of pi are given here though I understand that 50 billion digits (!) have been calculated already:

Pi = 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ...

Probably the most famous formula for determining pi is Leibnitz' formula=

Pi = 4 - (4/3) + (4/5) - (4/7) + (4/9) - (4/11) =

Summation (from n=0 to infinity) of [(-1)^n][4/(2n+1)]

Another famous summation involving pi was discovered by Euler as:

(Pi^2)/6 = 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 +.... + (1/n)^2

2) e --- The natural logarithm base, supposed named after the great mathmatician Leonhard Euler. The first hundred digits of e are given here as well:

e = 2.7182818284 5904523536 0287471352 6624977572 4709369995 9574966967 6277240766 3035354759 4571382178 5251664274 ...

I offer my students two ways to remember how to calculate the value of e:

e = limit (as n -> infinity) of (1 + 1/n)^n

e= Summation (from n=0 to infinity) of simply (1/n!)

3) Feigenbaum's Number --- This number, first shown by Becker and Dorfler, was demonstrated by Mitchell Feigenbaum to be a fundamental constant of nature having to do with the ratio of intervals of growth rate versus the doubling of up and down cycles characteristic of that rate. Keith Briggs, a scientist from the University of Melbourne, Australia, has calculated the most precise Feigenbaum number to date:

F = 4. 6692016091 0299067185 3203820466 2016172581 8557747576 8632745651
3430041343 3021131473 7138689744 0239480138 17165984855 1898151344
0862714202 7932522312 4429888908 9085994493 5463236713 4115324817 1421994745 5644365823 7932020095 6105833057 5458617652 2220703854 1064674949 4284981453 3917262005 6875566595 2339875603 825637225

4) Square Root of two = 1. 41421 35623 73095 0488...

5) Square Root of three = 1. 73205 08075 68877 2935...

6) Square Root of five = 2. 23606 79774 99789 6964...

7) Square Root of pi = 1.77245 38509 05516 02729 8167... (also known as Gamma(.5))

8) Square Root of e = 1. 64872 12707 00128 1468...

9) The Golden Mean, phi = (1 + sqrt(5)) / 2 = 1.61803 39887 99894...

10) e ^ pi = 23. 14069 26327 79267 006...

11) pi ^ e = 22. 45915 77183 61045 47342 715...

12) e ^ e = 15. 15426

13) Euler's constant (usually given as lower case gamma) = .57721 56649 01532 86060 6512...

= limit (as n goes from 1 to infinity) of (1/n - ln n)

14) 1 radian = the number of degrees that are subtended when the length of a radius is traced along the circumference of a circle.

1 radian = 180 / pi = 57. 29577 95130 8232...

15) ln 2 = .69315... = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...

= Summation (from n = 1 to infinity) of (-1)^(n+1) * (1/n)

16) ln 10 = 2.30259...

There is a library of more obscure mathematical constants HERE.

Polyhedra

Polyhedra, the plural of polyhedron, are three-dimensional solid figures with many geometrical faces to them. There are five commonly known regular polyhedra, regular meaning all faces are congruent and all edges and angles are congruent. They are:

 Tetrahedron 4 faces equilateral triangle Hexahedron (Cube) 6 faces square Octahedron 8 faces equilateral triangle Dodecahedron 12 faces pentagon Icosahedron 20 faces equilateral triangle

There is information regarding formulas to find the volumes, surface areas, inscribed radii, and circumscribed radii of the above polyhedra HERE.

There are also the Archimedean solids, solid shapes whose faces are all regular polygons of two or more kinds, and whose vertices are all identical. There are 13 different kinds. Two (the snub cube and snub dodecahedron) come in paired mirror-image forms. Eleven of these solids can be formed by truncating (chopping the corners off) simpler solids. They have pleasingly symmetrical crystalline shapes, and are described below. These eleven are:

 Truncated Tetrahedron 8 faces (4 triangles, 4 hexagons) Truncated Cube 14 faces (8 triangles, 6 octagons) Truncated Octahedron 14 faces (6 squares, 8 hexagons) Cuboctahedron 14 faces (8 triangles, 6 squares) Truncated Dodecahedron 32 faces (20 triangles, 12 dodecagons) - soccer ball pattern Truncated Icosahedron 32 faces (12 pentagons, 20 hexagons) - soccer ball / fullerene shape

 Icosidodecahedron 32 faces (20 triangles, 12 pentagons) Small Rhombicuboctahedron 26 faces (8 triangles, 18 squares) Great Rhombicuboctahedron 26 faces (12 squares, 8 hexagons, 6 octagons) Small Rhombicosidodecahedron 62 faces (20 triangles, 30 squares, 12 pentagons) Great Rhombicosidodecahedron 62 faces (30 squares, 20 hexagons, 12 dodecagons)

Thanks to Grant Hutchison for the info.

Spherical Harmonics

Spherical harmonics are expressions in three-dimensional spherical coordinates which are primarily used to describe the theoretical hybrid electron orbital shapes in molecules. The three coordinates are r (for radius), theta (degrees in the traditional x-y plane), and phi (degrees in the y-z plane). You may also recognize this way of laying out spatial coordinates from Star Trek's "210 mark 45" designation for navigation as the degrees in theta and phi. As with the rectangular coordinates, x, y, and z, we can describe any point in three dimensional space using such a coordinate system. All types of scientists use spherical and cylindrical (rho, theta, and z) coordinate systems to analyze various physical phenomena.

Here is the general formula used to produce these mathematical "flying saucers" with some examples below...

rho = sin(a * phi)^b + cos(c * phi)^d + sin(e * theta)^f + cos(g * theta)^h

r = (cos (theta))^2 + (cos(2 * theta))^4 + sin(4 * phi)

r = (cos(12 * theta))^5 + (cos(8 * theta))^3 + cos(6 * theta)

r = 2 * (cos(6 * theta))^6 - 4 * (cos(4 * theta))^4 - 2 * (cos(2 * theta))^2

rho = (sin(theta))^4 + (sin(2 * theta))^2 + e ^ (1 - sin(z))

rho = 4 * (cos(4 * theta))^4 - 2 * (cos(2 * theta))^2 + (1 + cos (z))^2

You can experiment with an infinite number of possibilities. You will soon discover what each coefficient, exponent, and function does to the overall shape of the object. Happy Hunting!

For a very large and extended study of over 640 images, go HERE.