Principal value of complex number

By convention the positive real axis is drawn pointing rightward, the positive imaginary axis is drawn pointing upward, principal value of complex number 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 magnitudefor the modulus, and phase[3] [1] for the argument, are sometimes used equivalently.

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.

Principal value of complex number

In mathematics , specifically complex analysis , the principal values of a multivalued function are the values along one chosen branch of that function , so 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. 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. However, there are other solutions, which is evidenced by considering the position of i in the complex plane and in particular its argument arg 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. Each value of k determines what is known as a branch or sheet , a single-valued component of the multiple-valued log function. In general, if f z is multiple-valued, the principal branch of f is denoted. Complex valued elementary functions can be multiple-valued over some domains. 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.

Glasgow: HarperCollins. The principal value of complex number argument measured in radians can be defined as:.

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 plays an important role in Mathematics. It is also known as z-plane, which is composed of two mutually perpendicular lines called axes. The horizontal line represents real numbers and is known as the real axis. On the other hand, the vertical line denotes imaginary numbers and is termed as an imaginary axis. The complex plane is used to represent a geometric interpretation of complex numbers.

Principal value of complex number

In mathematics , specifically complex analysis , the principal values of a multivalued function are the values along one chosen branch of that function , so 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. 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. 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.

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Navigation Navigation Add a new article Search in all topics Search in namespaces Search in categories Search using prefix. However, there are other solutions, which is evidenced by considering the position of i in the complex plane and in particular its argument arg i. Because it's defined in terms of roots , it also inherits the principal branch of square root as its own principal branch. ISBN X. Categories Categories Trigonometry Complex analysis Signal processing. More Articles for Maths. Glasgow: HarperCollins. We hope that the above article is helpful for your understanding and exam preparations. The argument calculated in the above step has certain ambiguity. Retrieved The complex plane is similar to the cartesian plane and illustrates a geometric interpretation of complex numbers. The principal value of complex number argument measured in radians can be defined as:. This represents an angle of up to half a complete circle from the positive real axis in either direction. 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.

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. An example of this is shown in the figure below. Note as well that we can now get a geometric interpretation of the modulus.

Beardon, Alan The argument of a complex no. Less than. Case 4. If you are checking Argument of Complex Number article, also check the related maths articles in the table below:. When any real-valued angle is considered, the argument is a multivalued function operating on the nonzero complex numbers. Tools Tools. If z 1 and z 2 are two non-zero complex numbers, then. Read Edit View history. One of the main motivations for defining the principal value Arg is to be able to write complex numbers in modulus-argument form. Some further identities follow. Report An Error. Categories Categories Trigonometry Complex analysis Signal processing.