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.

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.

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. 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. Note that notation varies, so arg and Arg may be interchanged in different texts. The set of all possible values of the argument can be written in terms of Arg as:. If a complex number is known in terms of its real and imaginary parts, then the function that calculates the principal value Arg is called the two-argument arctangent function, atan2 :. One of the main motivations for defining the principal value Arg is to be able to write complex numbers in modulus-argument form. Hence for any complex number z ,. Some further identities follow. If z 1 and z 2 are two non-zero complex numbers, then.

Ahlfors, Lars Complex valued elementary functions can be multiple-valued over some domains.

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.

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

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. We will therefore only consider the polar form of non-zero complex numbers.

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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. The argument calculated in the above step has certain ambiguity. The argument of Z is the angle drawn from the positive axis to the line segment. Category : Complex analysis. Let us now understand the formula and condition for the argument of complex numbers in different quadrants. Glasgow: HarperCollins. View Test Series. Learn the various Operations of Complex Numbers here. The argument of a certain number raised to some power is equivalent to the power multiplied by the argument. Formula for Argument of Complex Numbers Consider the below figure, for the complex no. Last updated on May 5, As we are taking the imaginary part, any normalisation by a real scalar will not affect the result.

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.

Beardon, Alan ISBN X. Values along one branch of a multivalued function so that it is single-valued. Short description : Angle of complex number about real axis. Download as PDF Printable version. Ahlfors, Lars For the line segment, OZ is the modulus of the complex number. Now, for example, say we wish to find log i. Collins Dictionary 2nd ed. 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. Page tools Page tools.