Complex numbers - Complex numbers From Physics Notes The...

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Complex numbers From Physics Notes The imaginary unit , denoted , is a hypothetical solution to the quadratic equation which is an equation that lacks real solutions. In other words, . We can let the imaginary unit take part in the usual arithmetic operations of addition and multiplication, treating it as an algebraic quantity on the same footing as the more familiar real numbers. Thus, we deal with numbers containing both real and imaginary parts, called complex numbers . It is one of the most profound discoveries of mathematics that this seemingly arbitrary idea gives rise to powerful computational methods for addressing mathematical and physical problems. Contents 1 Real and imaginary parts 2 Magnitudes and conjugates 3 Euler's formula 4 Polar representation 5 The complex plane 6 Complex functions 6.1 Complex trigonometric functions 6.2 Complex trigonometric identities 6.3 Hyperbolic functions 7 Trajectories in the complex plane 8 Exercises Real and imaginary parts For any complex number , we can write where and are real numbers that depend uniquely on . We refer to these as the real and imaginary parts of , and denote them as and . The and operations can be regarded as functions mapping complex numbers to real numbers The set of complex numbers is denoted by . We can define algebraic operations on complex numbers— addition/subtraction, products, and taking powers—simply by following the usual rules of algebra and setting whenever it shows up. Let , where . What is ?
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Example There's one exception: for now, we'll only consider taking integer powers, such as or . Taking non-integer powers, such as , introduces vexatious complications which we'll postpone for now (until the article on branch points and branch cuts). Of particular interest is the fact that One of the important consequences of this "linearity" feature is that if we have a complex function of a real variable, then we can calculate the derivative of that function from the derivatives of the real and imaginary parts: Example If is a complex function of a real input , then This is easily proven using the definition of the derivative: The case works out similarly. Note that the infinitesimal quantity is real; otherwise, this wouldn't work. Magnitudes and conjugates For a complex number , the magnitude of the complex number is
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This is a non-negative real number. We can show that complex magnitudes have the property This property is similar to the "absolute value" operation for real numbers, hence the similar notation. In particular, taking a power of a complex number raises its magnitude by the same power: We can define the complex conjugate of the complex number as . A complex number and its conjugate have the same magnitude: . We can show that complex conjugation obeys the properties Example Let us prove that . Let and . Then: Euler's formula Euler's formula is an extremely important result which states that This can be proven using the series definition of the exponential function:
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