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.

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.

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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.

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

Principal value of complex number

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.

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This article needs additional citations for verification. 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. What is the formula for modulus of a complex number? This represents an angle of up to half a complete circle from the positive real axis in either direction. Argument of Complex Number Solved Examples Having read about the formula, related properties and how to find the argument of a given complex number; let us practise some solved examples to understand the same. Similarly, from the periodicity of sin and cos , the second definition also has this property. For the use of the term principal value in describing improper integrals, see Cauchy principal value. ISBN Ahlfors, Lars 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.

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.

Categories : Trigonometry Complex analysis Signal processing. Ponnuswamy, S. Foundations of Complex Analysis 2nd ed. In general, if f z is multiple-valued, the principal branch of f is denoted. Formula for Argument of Complex Numbers Consider the below figure, for the complex no. The argument of zero is usually left undefined. Privacy policy About HandWiki Disclaimers. It is defined as the complex number w such that. Using Arg z instead of arg z , we obtain the principal value of the logarithm, and we write [1]. Some further identities follow.