Principal value
Encyclopedia
In considering complex multiple-valued functions in complex analysis
, the principal values of a function are the values along one chosen branch of that function, so it is single-valued
.
function log z. It is defined as the complex number
w such that
Now, for example, say we wish to find log i. This means we want to solve
for w. Clearly iπ/2 is a solution. But is it the only solution?
Of course, there are other solutions, which is evidenced by considering the position of i in the Argand plane and in particular its argument arg i. We can rotate counterclockwise π/2 radians from 1 to reach i initially, but if we rotate further another 2π we reach i again. So, we can conclude that i(π/2 + 2π) is also a solution for log i. It becomes clear that we can add any multiple of 2πi to our initial solution to obtain all values for log i.
But this has a consequence that may be surprising in comparison of real valued functions: log i does not have one definite value! For log z, we have
for an integer k, where Arg z is the (principal) argument of z defined to lie in the interval
. Each value of k determines what is known a branch (or sheet), a single-valued component of the multiple-valued log function.
The branch corresponding to k=0 is known as the principal branch, and along this branch, the values the function takes are known as the principal values.
such that for z in the domain of f, pv f(z) is single-valued.
Now, arg z is intrinsically multivalued. One often defines the argument of some complex number to be between -π (exclusive) and π (inclusive), so we take this to be the principal value of the argument, and we write the argument function on this branch Arg z (with the leading capital A). Using Arg z instead of arg z, we obtain the principal value of the logarithm, and we write
?
Consider
with 
. One usually defines zα to be eα log z. Yet eα log z is multiple-valued since we are using log as opposed to Log. Using Log we obtain the principal value of zα, i.e.,
the principal value of the square root
is :
with argument
measured in radian
s can be defined as:
To compute these values one can use functions :
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, the principal values of a function are the values along one chosen branch of that function, so it is single-valued
Single-valued function
A single-valued function is an emphatic term for a mathematical function in the usual sense. That is, each element of the function's domain maps to a single, well-defined element of its range. This contrasts with a general binary relation, which can be viewed as being a multi-valued function...
.
Motivation
Consider the complex logarithmComplex logarithm
In complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the natural logarithm ln x is the inverse of the real exponential function ex. Thus, a logarithm of z is a complex number w such that ew = z. The notation for such a w is log z...
function log z. It is defined as the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
w such that
Now, for example, say we wish to find log i. This means we want to solve
for w. Clearly iπ/2 is a solution. But is it the only solution?
Of course, there are other solutions, which is evidenced by considering the position of i in the Argand plane and in particular its argument arg i. We can rotate counterclockwise π/2 radians from 1 to reach i initially, but if we rotate further another 2π we reach i again. So, we can conclude that i(π/2 + 2π) is also a solution for log i. It becomes clear that we can add any multiple of 2πi to our initial solution to obtain all values for log i.
But this has a consequence that may be surprising in comparison of real valued functions: log i does not have one definite value! For log z, we have
for an integer k, where Arg z is the (principal) argument of z defined to lie in the interval
. Each value of k determines what is known a branch (or sheet), a single-valued component of the multiple-valued log function.The branch corresponding to k=0 is known as the principal branch, and along this branch, the values the function takes are known as the principal values.
General case
In general, if f(z) is multiple-valued, the principal branch of f is denotedsuch that for z in the domain of f, pv f(z) is single-valued.
Principal values of standard functions
Complex valued elementary functions can be multiple valued over some domains. Determining the principal value of some of these functions can be obtained by decomposing the function into simpler ones whereby the principal value of the simple functions are straightforward to obtain.Logarithm function
We have examined the logarithm function above, i.e.,Now, arg z is intrinsically multivalued. One often defines the argument of some complex number to be between -π (exclusive) and π (inclusive), so we take this to be the principal value of the argument, and we write the argument function on this branch Arg z (with the leading capital A). Using Arg z instead of arg z, we obtain the principal value of the logarithm, and we write
Exponential function
So far we have only considered the logarithm function. What about exponentsExponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
?
Consider
with . One usually defines zα to be eα log z. Yet eα log z is multiple-valued since we are using log as opposed to Log. Using Log we obtain the principal value of zα, i.e.,
Square root
For a complex numberComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
the principal value of the square rootSquare root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
is :
with argument
Arg (mathematics)
In mathematics, arg is a function operating on complex numbers . It gives the angle between the line joining the point to the origin and the positive real axis, shown as in figure 1 opposite, known as an argument of the point In mathematics, arg is a function operating on complex numbers...
Complex argument
The principal value of complex number argumentArg (mathematics)
In mathematics, arg is a function operating on complex numbers . It gives the angle between the line joining the point to the origin and the positive real axis, shown as in figure 1 opposite, known as an argument of the point In mathematics, arg is a function operating on complex numbers...
measured in radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s can be defined as:
- values in the range [0, 2π)
- values in the range (-π, π].
To compute these values one can use functions :
- atan2Atan2In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive -axis of a plane and the point given by the coordinates on it...
with principal value in the range (-π, π] - atan with principal value in the range (-π/2, π/2]