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Complex Variables Demysti ed
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Since the exponential function is entire, the cosine and sin functions are also entire Other derivatives follow from elementary calculus: d tan z = sec 2 z dz d sec z = sec z tan z dz d cot z = csc 2 z dz d csc z = csc z cot z dz (454) (455)
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The derivatives of the hyperbolic functions can be derived easily using exponential representations: d cosh z = sinh z dz d tanh z = sech 2 z dz d sech z = sech z tanh z dz Finally, we note the derivative of a complex exponent: d z = z 1 dz (459) d sinh z = cosh z dz (456) (457) (458)
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Note, however, that since this is a multivalued function, this holds for z > 0, 0 < arg z < 2 or some other interval
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Branches
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A multivalued function repeats itself when z moves in a complete circle about the origin in the complex plane When 0 < 2 , the function is single valued We say that we are on one branch of the function But as we let z traverse the circle again so we enter the region where 2 < , the function repeats We say that we ve entered another branch of the function A multivalued function like this repeats itself any number of times For convenience, a barrier is set up at our choosing in the complex plane where we do not allow z to cross This barrier is called a branch cut The point from which the branch cut originates is called a branch point The branch cut extends out from the branch point to in nity For example, for a multivalued function, we can take the branch point to be the origin and the branch cut can extend out from the origin to positive in nity (Fig 416)
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Elementary Functions
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Figure 416 Some multivalued functions repeats themselves after z has completely gone around the origin We prevent the function from being multivalued by staying on one branch This means we cannot cross the branch cut, which we have chosen in this case to be the line from the origin to positive in nity Note that a circle does not have to be used, we just have to let z go completely around the origin a circle was used here for simplicity
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In this chapter, we described the basic properties of some elementary functions encountered in complex variables These included polynomials, the complex exponential, the trig functions, the logarithm, the hyperbolic functions, and functions with complex exponents
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Quiz
1 Prove that cos( x + iy) = cos x cos iy sin x sin iy 2 If f ( x ) = e x, then f can never be negative Is the same true of e z 3 Find a compact expression for e2+3 i 4 Find an identity for 1 + tan 2 z by using Eq (412) 5 Find an identity for tan( z + w) 6 Are the inverse trig functions multivalued
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Sequences and Series
It is common practice and often a necessity to represent a function of a real variable using an in nite series expansion It turns out that this is also true when working with complex functions As we will see, there are some new concepts involved when working with complex functions We begin by considering sequences
Sequences
Consider the positive integers n = 1, 2, 3, and consider a function on the positive integers, which we denote by f (n) We call such a function a sequence The output of the function is a number: f (n) = an So a sequence is an ordered set of numbers a1 , a2 , a3 , and we refer to an as the nth term in the sequence Sequences can also be indicated using curly braces, so we can write { f (n)} or {an }
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