Features: Basic Mathematics
The following table summarizes which types of input are handled
automatically by each of Mathematica CalcCenter's principal
arithmetic
and basic math functions.
| Function
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Real Input
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Complex Input
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Symbolic Input
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Automatic Unit Conversion
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Full-Range Input
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Plus
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Minus
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Times
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Divide
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Power
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Exp
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Log
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Sin
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ArcSin
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Sinh
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ArcSinh
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Cos
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ArcCos
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Cosh
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ArcCosh
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Sec
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ArcSec
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Sech
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ArcSech
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Csc
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ArcCsc
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Csch
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ArcCsch
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Tan
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ArcTan
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Tanh
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ArcTanh
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Cot
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ArcCot
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Coth
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ArcCoth
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BesselJ
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BesselK
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Gamma
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Erf
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InverseErf
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Fibonacci
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Factorial
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Binomial
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Mod
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Round
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UnitStep
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Abs
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Arg
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Re
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Im
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Conjugate
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ExpToTrig
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N/A
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TrigToExp
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N/A
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![[Graphics:Images/specifications_gr_32.gif]](Images/specifications_gr_32.gif)
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The following table lists the principal algorithms used in calculating
numerical results in Mathematica CalcCenter.
|
Case
|
Algorithm
|
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Reciprocals
|
Newton's method is used.
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Rational powers
|
Newton's method is used.
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Exponential and trigonometric functions
|
Taylor series, stable recursion by argument doubling, and functional
relations are used.
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Log and inverse trigonometric functions
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Taylor series and functional relations are used.
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Special functions
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Mathematica-derived rational minimax approximations are used.
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Fibonacci
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Iterative method based on the binary-digit sequence is used.
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Notes
 Real
Input: A function must accept positive and negative real numbers and
produce the appropriate result whether the result is real or
complex (e.g., Sin[-200] = 0.8732"9"7).
 Complex
Input: A function must accept input that contains any combination of
real and imaginary numbers, producing appropriate real or complex output
(e.g., Sin[4 - 20 I] = -1.83587 x 108 + 1.58563 x
108 I).
 Symbolic
Input: A function must accept any symbol or composite symbolic
expression, produce appropriate output (symbolic, real, or complex), and
be understood by symbolic operators such as Integrate and
Simplify (e.g.,  ).
 Automatic
Unit Conversion: A function must take arguments containing any units
and automatically establish the resulting unit (e.g., 20  + 100
 = 0.5091574
 ).
 Full-Range
Input is defined to mean that input within the defined range
of a function, and in the range  I
but not within  I, the function will
give at least five significant figures of correct result if sufficient
significant figures of input are provided.
Note that for particular values near the singularities of asymptotic
functions, rounding errors may be more significant. It may also be
possible in some circumstances to achieve much higher performance. For
example, Sin[1234567890.12345] uses input that is much greater
than  and still exceeds the required accuracy.
Full-range input functions must also accept
data sets with no limit to the number of data points in the set. Note
that there will always be practical limitations on memory or available
computational time required for operations on large data sets. There is no
limit to the size of data sets (1,000,000 integer data points require
approximately 8 MB of RAM, and sorting 1,000,000 points takes
approximately 0.36 seconds on a 3.6 GHz Pentium IV).
Range depends on relative values of input arguments.
Accurate results are given over input range 
Accurate results are given over input range 
InverseErf is defined only on the interval [-1, 1].
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![[ja]](/common/images2003/lang_bottom_ja.gif)