Principal value of complex number

In mathematicsspecifically complex analysisthe principal values of a multivalued function are the values along one chosen branch of that functionso that it is single-valued. A simple case arises in taking the square root of a positive real number. Consider the complex logarithm function log z, principal value of complex number.

By convention the positive real axis is drawn pointing rightward, the positive imaginary axis is drawn pointing upward, and complex numbers with positive real part are considered to have an anticlockwise argument with positive sign. When any real-valued angle is considered, the argument is a multivalued function operating on the nonzero complex numbers. The names magnitude , for the modulus, and phase , [3] [1] for the argument, are sometimes used equivalently. Similarly, from the periodicity of sin and cos , the second definition also has this property. The argument of zero is usually left undefined. Because it's defined in terms of roots , it also inherits the principal branch of square root as its own principal branch. This represents an angle of up to half a complete circle from the positive real axis in either direction.

Principal value of complex number

A complex number is an important section of mathematics as it is the combination of both real and imaginary elements. In the graphical representation, the horizontal line is used for the real numbers and the vertical lines is used to plot the imaginary numbers. Two concepts that come into the picture with the graphical representation of complex no. The modulus in mathematics is the square root of the summation of the squares of the real part plus the imaginary part of the complex number. On the other hand, the argument is the angle created with the positive direction of the real axis. With this article, we will learn about the argument of complex number formulas with the definition, solved examples and properties. The complex plane is similar to the cartesian plane and illustrates a geometric interpretation of complex numbers. Herein, the real part is on the real axis i. The argument of a complex number is the inclined angle developed in between the real axis and the complex number in the direction of the complex no. Check out the below image to understand the same. Learn more about the polar form of complex numbers. Consider the below figure, for the complex no. For the line segment, OZ is the modulus of the complex number. The argument of Z is the angle drawn from the positive axis to the line segment.

For the theorem, see Argument principle. Namespaces Page Discussion.

By convention the positive real axis is drawn pointing rightward, the positive imaginary axis is drawn pointing upward, and complex numbers with positive real part are considered to have an anticlockwise argument with positive sign. When any real-valued angle is considered, the argument is a multivalued function operating on the nonzero complex numbers. The names magnitude , for the modulus, and phase , [3] [1] for the argument, are sometimes used equivalently. Similarly, from the periodicity of sin and cos , the second definition also has this property. The argument of zero is usually left undefined. Because it's defined in terms of roots , it also inherits the principal branch of square root as its own principal branch. This represents an angle of up to half a complete circle from the positive real axis in either direction.

A multiple-valued function can be considered as a collection of single-valued functions, each member of which is called a branch of the function. In general, we consider one particular member as a principal branch of the multiple-valued function and the value of the function corresponding to this branch as the principal value. Hence, the function. In a similar fashion, the complex logarithm is a complex extension of the usual real natural i. The other four trigonometric functions are defined in terms of the sine and cosine functions with the following relations:.

Principal value of complex number

The imaginary unit number is used to express the complex numbers, where i is defined as imaginary or unit imaginary. We will explain here imaginary numbers rules and chart, which are used in Mathematical calculations. The basic arithmetic operations on complex numbers can be done by calculators. The imaginary number i is also expressed as j sometimes. Basically the value of imaginary i is generated, when there is a negative number inside the square root, such that the square of an imaginary number is equal to the root of But when we take the cube of i, the value is -i. The imaginary number, when multiplied by itself, gives a negative value. For example, consider an imaginary number 3 i , if multiplied by itself or if we take the square of 3 i , gives 9 i 2 or we can write it as Also, 0 is considered as both the real number and an imaginary number. Basically, first, it was considered as there are no such facts as the imaginary number and real numbers.

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If you are checking Argument of Complex Number article, also check the related maths articles in the table below:. What is the formula for modulus of a complex number? Hidden categories: Articles with short description Short description matches Wikidata Articles needing additional references from March All articles needing additional references. The principal value sometimes has the initial letter capitalized, as in Arg z , especially when a general version of the argument is also being considered. ISBN X. Because it's defined in terms of roots , it also inherits the principal branch of square root as its own principal branch. Now, for example, say we wish to find log i. Some further identities follow. The argument of Z is the angle drawn from the positive axis to the line segment. View Test Series. Beardon, Alan

Before we get into the alternate forms we should first take a very brief look at a natural geometric interpretation of complex numbers since this will lead us into our first alternate form.

As we are taking the imaginary part, any normalisation by a real scalar will not affect the result. The related formula is as follows:. Some of the important argument properties of complex numbers are as follows:. Read more. Complex valued elementary functions can be multiple-valued over some domains. Wiki tools Wiki tools Special pages Cite this page. To determine the arg z value, fetch the real and imaginary components and substitute the values in the formula. By convention the positive real axis is drawn pointing rightward, the positive imaginary axis is drawn pointing upward, and complex numbers with positive real part are considered to have an anticlockwise argument with positive sign. For the line segment, OZ is the modulus of the complex number. Note that notation varies, so arg and Arg may be interchanged in different texts. Borowski, Ephraim; Borwein, Jonathan 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. Category : Complex analysis.