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Appendix
A.1 Pure Functions
A function is a correspondence between two sets of numbers A and B such that for each number in A there corresponds a unique number in B. For example, the squaring function: For each real number there corresponds a unique non-negative real number called its square. While it is customary to write f(x) = x2, one must understand that there is no special significance to the letter x. It is the process of squaring that defines the function. Although Mathematica allows a function to be defined in terms of a variable, as in f[x_]= x2, the variable x acts as a dummy and is insignificant. The function would be the same had we used y, z, or any other symbol. A pure function is defined without reference to any specific variable. Its arguments are labeled #1, #2, #3, and so forth. To distinguish a pure function from any other Mathematica construct, an ampersand, &, is used at the end of its definition. Once defined, we can deal with a pure function as we would any other function. Although the concept of a pure function is a natural one, it is possible to use Mathematica and never be concerned with it. Occasionally, however, Mathematica will express an answer as a pure function and it is therefore worthy of a brief mention. The interested reader can find more information in Mathematica s Documentation Center.
EXAMPLE 1
f = #12 &; f[3] 9 f[x] x2 f[a + b] (a + b)2
EXAMPLE 2
g = #1 #22 + 3&; g[3, 4] 51 g[u, v] 3 + u v2
APPENDIX
Another way of specifying a pure function is by use of Mathematica s Function command.
Function[x, body]is a pure function with a single parameter x. Function[{x1, x2,...}, body] is a pure function with a list of parameters x1, x2,...
EXAMPLE 3 Express the solution of the differential equation
d2y + y = 0; y '(0) = y(0) = 1 dx 2 as a pure function and evaluate it for x = /4. DSolve[{y''[x]+ y[x] 0, y[0] 1, y [0]= 1}, y, x]
{{y
Function[{x}, Cos[x] + Sin[x]]}} Function[{x}, Cos[x]+ Sin[x]][o/4]
SOLVED PROBLEMS
A.1 Express as a pure function the process of adding the square of a number to its square root and compute its value at 9.
SOLUTION
f = #1^2 + Sqrt[#1] &; f[9] 84
A.2 A number is formed from two other numbers by adding the square of their sum to the sum of their squares. Express this operation as a pure function and compute its value for the numbers 3 and 4.
SOLUTION
g =(#1 + #2)^2 + #1^2 + #2^2 &; g[3, 4] 74
A.3 Express the derivative of the function Sin as a pure function and compute its vale at /6.
SOLUTION
f = Sin' Cos[#1]& f[o/6] 3 2
A.4 Define f (x) = (1 + x + x2)5 and express its second derivative as a pure function.
SOLUTION
f[x_] = (1 + x + x2)5; f''
4 20(1 + 2 #1)2 (1 + #1 + #12)3 + 10(1 + #1 + #12) &
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