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EXAMPLE 17 How long does it take the kernel to compute the ten billionth prime
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Timing[Prime[10 000 000 000]] {2.953,252 097 800 623} Of course, the actual time taken will vary, depending upon the speed of the CPU.
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Logarithms and exponential functions to any base can be computed using the function Log.
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Log[x] represents the natural logarithm. If a base, b, other than e is required, the appropriate form is Log[b, x]. The function Exp[x] is the natural exponential function. Other equivalent forms are E^x and Ex. Lowercase e cannot be used, but the special symbol from the Basic Math Input palette may be used instead. Exponential functions to the base b are computed by b^x or bx.
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EXAMPLE 18 Compute ln 100, the natural logarithm of 100.
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Log[100] Log[100] Log[100]//N 4.60517
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Observe that Mathematica always gives exact answers. Approximations are supplied only when requested.
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EXAMPLE 19 Compute log2100.
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Log[2, 100] Log[100] Log[2] Log[2, 100]//N 6.64386
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EXAMPLE 20 To compute a numerical approximation of e2, we can write
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This is the exact value of log2100, expressed in terms of natural logarithms.
Exp[2]//N 7.38906
E2//N
2//N
The six basic trigonometric functions, sine, cosine, tangent, secant, cosecant, and cotangent, are represented in Mathematica by Sin, Cos, Tan, Sec, Csc, and Cot, respectively.
Mathematica assumes the arguments of trigonometric functions to be in radians. Problems involving degrees must first be converted to radians if trigonometric functions are involved. For this purpose, one can use the built-in constant, Degree, whose value is /180. The symbol , located on the Basic Math Input palette, may be used as well.
EXAMPLE 21 60 is equivalent to /3 radians. To compute its sin using radian measure, we write
Sin 3 3 2
Sin[Pi/3]
Basic Concepts
If we wish to compute its sin using degree measure, we can type
Sin[60 Degree] 3 2 or Sin[60 ]
Care must be taken with trigonometric powers. The square of sin x in trigonometric form is traditionally 2 written sin 2 x , but Mathematica will accept only Sin[x] or Sin[x]^2.
EXAMPLE 22 Compute the square of sin 60 .
Sin[60o]2 or 3 4
Sin[60 Degree]^2
The inverse trigonometric functions are ArcSin, ArcCos, ArcTan, ArcSec, ArcCsc, and ArcCot. However only the principal values, expressed in radians, are returned by these functions.
EXAMPLE 23
ArcSin[1] 2 ArcCos[Cos[3 ]]
Cos[3o]= 1 but the principal value of ArcCos[ 1] is o.
Hyperbolic functions are combinations of exponential functions which have interesting mathematical properties. There are six hyperbolic functions. The three basic ones are sinh x = e x e x 2 cosh x = e x + e x 2 tanh x = e x e x e x + e x
The other three, sech x, csch x, and coth x, are reciprocals, respectively, of cosh x, sinh x, and tanh x.
The Mathematica representations of the six hyperbolic functions are Sinh, Cosh, Tanh, Sech, Csch, and Coth.
EXAMPLE 24 Compute a numerical approximation to sinh 2.
Sinh[2]//N 3.62686
The inverse hyperbolic functions are represented by ArcSinh, ArcCosh, ArcTanh, ArcSech, ArcCsch, and ArcCoth.
Because Cosh and Sech are not one-to-one, ArcCosh and ArcSech return only positive values for real arguments.
EXAMPLE 25
ArcSinh[ 2] //N 1.44364 ArcCosh[2] //N 1.31696
One special command is worthy of mention at this time:
Print[expression] prints expression, followed by a line feed. Print[expression1, expression2, . . . ] prints expression1, expression2, . . . followed by a single line feed.
Basic Concepts
At first glance it may seem that Print is a redundant command, as simply typing the name of any object will reveal its value. However, it has a useful purpose (e.g., see loops in Section 2.8).
EXAMPLE 26
Print["This prints a line of text."] This prints a line of text.
EXAMPLE 27
a = 1; b = 2; c = 3; d = 4; e = 5; Print[a + b, b + c, c + d, d + e, e + a] 35796
Mathematica includes a class of functions ending in the letter Q: AlgebraicIntegerQ
AlgebraicUnitQ ArgumentCountQ ArrayQ AtomQ CoprimeQ DigitQ DistributionDomainQ DistributionParameterQ EllipticNomeQ EvenQ ExactNumberQ FreeQ HermitianMatrixQ HypergeometricPFQ InexactNumberQ IntegerQ IntervalMemberQ InverseEllipticNomeQ
LegendreQ
LetterQ LinkConnectedQ LinkReadyQ ListQ LowerCaseQ MachineNumberQ MatchLocalNameQ MatchQ MatrixQ MemberQ NameQ NumberQ NumericQ OddQ OptionQ OrderedQ PartitionsQ PolynomialQ
PositiveDe niteMatrixQ
PossibleZeroQ PrimePowerQ PrimeQ QuadraticIrrationalQ RootOfUnityQ SameQ SquareFreeQ StringFreeQ StringMatchQ StringQ SyntaxQ TensorQ TrueQ UnsameQ UpperCaseQ ValueQ VectorQ
These functions are used to test for certain conditions and return a value of True or False. Their precise syntax can be determined from the Help menu or by using as illustrated in the next examples.
EXAMPLE 28
PrimeQ
PrimeQ[expr] yields True if expr is a prime number, and yields False otherwise.
PrimeQ[5] True PrimeQ[6] False
EXAMPLE 29
PolynomialQ
PolynomialQ[expr, var] yields True if expr is a polynomial in var, and yields False otherwise. PolynomialQ[expr, {var1,...}] tests whether expr is a polynomial in the vari.
Basic Concepts
PolynomialQ[x2 y + x + True PolynomialQ[x2 y + x + False
y, x]
y, y]
SOLVED PROBLEMS
2.5 Compute numerical approximations to the square root and cube root of 10.
SOLUTION
10 //N 3.16228
or or
Sqrt[10] //N 10^(1/3) //N
10 //N
2.6 Compute numerical approximations to the square root and cube root of 10 accurate to 20 significant digits.