Standard functions

6.16.2 Standard functions

The max command finds the maximum of a sequence of real numbers.
- max takes an arbitrary number of arguments:
  seq, a sequence (or list) of real numbers.
- max(seq) returns the largest number in the sequence seq.
Example.
Input:
max(0,1,2,-1,-2)
Output:
2
The min command finds the minimum of a sequence of real numbers.
- min takes an arbitrary number of arguments:
  seq, a sequence (or list) of real numbers.
- min(seq) returns the smallest number in the sequence seq.
Example.
Input:
min(0,1,2,-1,-2)
Output:
−2
The abs command finds the absolute value of a real or complex number.
- abs takes one argument:
  x, a real or complex number.
- abs(x) returns the absolute value of x.
Examples.
- Input:
  abs(-5)
  Output:
  5
- Input:
  abs(3+4*i)
  Output:
  5
The sign command finds the sign of a real number (+1 if it is positive, 0 if it is zero, and -1 if it is negative).
- sign takes one argument:
  x, a real number.
- sign(x) returns the sign of x.
Examples.
- Input:
  sign(-4)
  Output:
  −1
- Input:
  sign(0)
  Output:
  0
The floor command finds the floor of a real number; namely, the largest integer less than or equal to the number.
iPart) is a synonym for floor.
- floor takes one argument:
  x, a real number.
- floor(x) returns the floor of x.
Examples.
- Input:
  floor(4.1)
  Output:
  4
- Input:
  floor(-4.1)
  Output:
  −5
The round command rounds a number to the nearest integer, rounding up in the case of a half-integer.
- round takes one argument:
  x, a real number.
- round(x) returns the nearest integer to x.
Examples.
- Input:
  round(3.4)
  Output:
  3
- Input:
  round(-3.4)
  Output:
  −3
- Input:
  round(3.5)
  Output:
  4
The ceil command finds the ceiling of a real number; namely, the smallest integer greater than or equal to the number.
ceiling is a synonym for ceil.
- ceil takes one argument:
  x, a real number.
- ceil(x) returns the ceiling of x.
Examples.
- Input:
  ceiling(4.1)
  Output:
  5
- Input:
  ceiling(-4.1)
  Output:
  −4
The frac command finds the fractional part of a number; informally, the part of the number to the right of the decimal point with the appropriate plus or minus sign. For a positive real number x, the fractional part is x minus the floor of x; for a negative real number x, the fractional part is x minus the ceiling of x.
fPart is a synonym for frac.
- frac takes one argument:
  x, a real number.
- frac(x) returns the fractional part of x.
Examples.
- Input:
  frac(3.24)
  Output:
  0.24
- Input:
  frac(-3.24)
  Output:
  −0.24
The trunc command truncates a real number; namely, it removes the fractional part. The truncated number added to the fractional part will equal the original number.
- trunc takes one argument:
  x, a real number.
- trunc(x) returns the truncated value of x.
Examples.
- Input:
  trunc(3.24)
  Output:
  3
- Input:
  trunc(-3.24)
  Output:
  −3
The id command is the identity function.
- id takes one argument or a sequence of arguments:
  seq, whose elements can be any type.
- id(seq) returns seq.
Example.
Input:
id(a,1,"abc",[1,2,3])
Output:
a,1,"abc", ⎡
⎣ 1,2,3 ⎤
⎦
The sq command squares its argument.
- sq takes one argument:
  x, any object that can be multiplied by itself.
- sq(x) returns x².
Examples.
- Input:
  sq(5)
  Output:
  25
- Input:
  sq(x+y)
  Output:
  ⎛
  ⎝ x+y ⎞
  ⎠ ²
- Input:
  sq([[1,2],[3,4]])
  (This is a matrix product; see Section 6.44).
  Output:
     ⎡
  ⎢
  ⎣
          7 10
  
          15 22
  
  ⎤
  ⎥
  ⎦
- Input:
  sq([1,2,3])
  (This is the dot product of [1,2,3] with itself.)
  Output:
  14
The sqrt command finds the square root of its argument.
- sqrt takes one argument:
  x, any object for which the 1/2 power makes sense.
- sqrt(x) returns x^1/2.
Examples.
- Input:
  sqrt(9)
  Output:
  3
- Input:
  sqrt((x+y)^2)
  Output:
  ⎪
  ⎪ x+y ⎪
  ⎪
- Input:
  simplify(sqrt([[2,3],[3,5]]))
  Output:
     ⎡
  ⎢
  ⎣
          1 1
  
          1 2
  
  ⎤
  ⎥
  ⎦
The surd command finds roots of quantities.
- surd takes two arguments:
  x and n, numbers.
- surd(x,n) returns the nth root of x; i.e., x^1/n.
Example.
Input:
surd(15.625,3)
Output:
2.5
The exp command computes the exponential function.
- exp takes one argument:
  x, a number.
- exp(x) returns e^x.
Example.
Input:
exp(1.0)
Output:
2.71828182846
The log command computes the natural logarithm function.
ln is a synonym for log.
- log takes one argument:
  x, a number.
- log(x) returns the natural logarithm of x.
Example.
Input:
log(2.0)
Output:
0.69314718056
The log10 computes the the base-10 logarithm.
- log10 takes one argument:
  x, a number.
- log10(x) returns the base-10 logarithm of x.
Example.
Input:
log10(1000)
Output:
3
The logb computes the logarithm to a specified base.
- logb takes two arguments:
  x and b, non-zero numbers.
- logb(x,b) returns the base-b logarithm of x.
Example.
Input:
logb(10.0,2)
Output:
3.32192809489
The standard trigonometric functions:
- The sin command is the sine function.
- The cos command is the cosine function.
- The tan command is the tangent function (tan(x)= sin(x)/cos(x)).
- The cot command is the cotangent function (cot(x)= cos(x)/sin(x)).
- The sec command is the secant function (sec(x)= 1/cos(x)).
- The csc command is the cosecant function (csc(x) = 1/sin(x)).
- These commands take one argument: x, a number.
  The number x will by default represent an angle measured in radians, but you can set Xcas to use degrees (see Section 3.5.3) by setting the variable angle_radian to 0; resetting it to 1 will change the angle measure to radians again.
- sin(x) returns the sine of x.
  
  Examples.
  - Input (with angle_radian equal to 1):
    sin(pi/4)
    Output:
    
    √
    
    2
    
    2
  - Input (with angle_radian equal to 0):
    sin(30)
    Output:
    
    1
    
    2
- cos(x) returns the cosine of x.
  
  Examples.
  - Input (with angle_radian equal to 1):
    cos(pi/6)
    Output:
    
    √
    
    3
    
    2
  - Input (with angle_radian equal to 0):
    cos(90)
    Output:
    0
- tan(x) returns the tangent of x.
  
  Examples.
  - Input (with angle_radian equal to 1):
    tan(pi/4)
    Output:
    1
  - Input (with angle_radian equal to 0):
    tan(60)
    Output:
    √
    
    3
- cot(x) returns the cotangent of x.
  
  Examples.
  - Input (with angle_radian equal to 1):
    cot(pi/6)
    Output:
    
    2 √
    
    3
    
    2
  - Input (with angle_radian equal to 0):
    cot(45)
    Output:
    1
- sec(x) returns the secant of x.
  
  Example.
  - Input (with angle_radian equal to 1):
    sec(pi/3)
    Output:
    2
  - Input (with angle_radian equal to 0):
    sec(30)
    Output:
    
    2
    
    √
    
    3
- csc(x) returns the cosecant of x.
  Examples.
  - Input (with angle_radian equal to 1):
    csc(pi/4)
    Output:
    
    2
    
    √
    
    2
  - Input (with angle_radian equal to 0):
    csc(30)
    Output:
    2
The asin, acos, atan, acot, asec, acsc commands are the inverse trigonometric functions. The latter are defined by:
- asec(x) = acos(1/x),
- acsc(x) = asin(1/x),
- acot(x) = atan(1/x).
arcsin is a synonym for asin.
arccos is a synonym for acos.
arctan is a synonym for atan.
- These functions take one argument: x, a number.
  They return a number which can represent an angle; by default, the angles will be in radians, but you can set Xcas to use degrees (see Section 3.5.3) by setting the variable angle_radian to 0; resetting it to 1 will change the angle measure to radians again.
- asin(x) returns the arcsine of x.
  
  Examples.
  - Input (with angle_radian equal to 1):
    asin(1/2)
    Output:
    
    π
    
    6
  - Input (with angle_radian equal to 0):
    asin(1)
    Output:
    
    π
    
    2
- acos(x) returns the arccosine of x.
  
  Examples.
  - Input (with angle_radian equal to 1):
    acos(sqrt(3)/2)
    Output:
    
    1
    
    6
    
    π
  - Input (with angle_radian equal to 0):
    acos(-1/2)
    Output:
    120
- atan(x) returns the arctangent of x.
  
  Examples.
  - Input (with angle_radian equal to 1):
    atan(sqrt(3))
    Output:
    
    π
    
    3
  - Input (with angle_radian equal to 1):
    atan(1)
    Output:
    45
- acot(x) returns the arccotangent of x.
  
  Examples.
  - Input (with angle_radian equal to 1):
    acot(sqrt(3))
    Output:
    
    π
    
    6
  - Input (with angle_radian equal to 0):
    acot(1/sqrt(3))
    Output:
    60
- asec(x) returns the arcsecant of x.
  
  Examples.
  - Input (with angle_radian equal to 1):
    asec(1)
    Output:
    0
  - Input (with angle_radian equal to 0):
    asec(sqrt(2))
    Output:
    45
- acsc(x) returns the arccosecant of x.
  
  Examples.
  - Input (with angle_radian equal to 1):
    acsc(1)
    Output:
    
    π
    
    2
  - Input (with angle_radian equal to 0):
    acsc(2)
    Output:
    30
The sinh, cosh, and tanh commands compute the hyperbolic sine, hyperbolic cosine, and hyperbolic tangent functions.
- These functions take one argument:
  x, a number.
- sinh(x) returns the hyperbolic sine of x.
  
  Example.
  Input:
  sinh(1.0)
  Output:
  1.17520119364
- cosh(x) returns the hyperbolic cosine of x.
  
  Example.
  Input:
  cosh(0)
  Output:
  1
- tanh(x) returns the hyperbolic tangent of x.
  
  Example.
  Input:
  tanh(-1.0)
  Output:
  −0.761594155956
The asinh, acosh, and atanh commands compute the inverse hyperbolic functions.
arcsinh is a synonym for asinh.
arccosh is a synonym for acosh.
arctanh is a synonym for atanh.
- These functions take one argument:
  x, a number.
- asinh(x) returns the inverse hyperbolic sine of x.
  
  Example.
  Input:
  asinh(2)
  Output:
  ln ⎛
  ⎜
  ⎝ 2+ √
  
  5
  
  ⎞
  ⎟
  ⎠
- acosh(x) returns the inverse hyperbolic cosine of x.
  
  Example.
  Input:
  acosh(1)
  Output:
  0
- atanh(x) returns the inverse hyperbolic tangent of x.
  
  Example.
  Input:
  atanh(1/2)
  Output:
  
  ln ⎛
  ⎝ 3 ⎞
  ⎠
  
  2