Before we start on the gallery we define the term entire function Definition A

Before we start on the gallery we define the term

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Before we start on the gallery we define the term “entire function”. Definition. A function that is analytic at every point in the complex plane is called an entire function . We will see that e z , z n , sin( z ) are all entire functions. 2.8.1 Gallery of functions, derivatives and properties The following is a concise list of a number of functions and their derivatives. None of the derivatives will surprise you. We also give important properties for some of the functions. The proofs for each follow below. 1. f ( z ) = e z = e x cos( y ) + i e x sin( y ). Domain = all of C ( f is entire). f 0 ( z ) = e z . 2. f ( z ) c (constant) Domain = all of C ( f is entire). f 0 ( z ) = 0 . 3. f ( z ) = z n ( n an integer 0) Domain = all of C ( f is entire). f 0 ( z ) = nz n - 1 . 4. P ( z ) (polynomial) A polynomial has the form P ( z ) = a n z n + a n - 1 z n - 1 + . . . + a 0 . Domain = all of C ( P ( z ) is entire). P 0 ( z ) = na n z n - 1 + ( n - 1) a n - 1 z n - 1 + . . . + 2 a 2 z + a 1 . 5. f ( z ) = 1 /z Domain = C - { 0 } ( the punctured plane ). f 0 ( z ) = - 1 /z 2 . 6. f ( z ) = P ( z ) /Q ( z ) (rational function). When P and Q are polynomials P ( z ) /Q ( z ) is called a rational function. If we assume that P and Q have no common roots, then: Domain = C - { roots of Q } f 0 ( z ) = P 0 Q - PQ 0 Q 2 .
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2 ANALYTIC FUNCTIONS 13 7. sin( z ), cos( z ) Definition. cos( z ) = e iz + e - iz 2 , sin( z ) = e iz - e - iz 2 i (By Euler’s formula we know this is consistent with cos( x ) and sin( x ) when z = x is real.) Domain: these functions are entire. d cos( z ) dz = - sin( z ) , d sin( z ) dz = cos( z ) . Other key properties of sin and cos: - cos 2 ( z ) + sin 2 ( z ) = 1 - e z = cos( z ) + i sin( z ) - Periodic in x with period 2 π , e.g. sin( x + 2 π + iy ) = sin( x + iy ). - They are not bounded! - In the form f ( z ) = u ( x, y ) + iv ( x, y ) we have cos( z ) = cos( x ) cosh( y ) + i sin( x ) sinh( y ) sin( z ) = sin( x ) cosh( y ) + i cos( x ) sinh( y ) (cosh and sinh are defined below.) - The zeros of sin( z ) are z = for n any integer. The zeros of cos( z ) are z = π/ 2 + for n any integer. (That is, they have only real zeros that you learned about in your trig. class.) 8. Other trig functions cot( z ), sec( z ) etc. Definition. The same as for the real versions of these function, e.g. cot( z ) = cos( z ) / sin( z ), sec( z ) = 1 / cos( z ). Domain: The entire plane minus the zeros of the denominator. Derivative: Compute using the quotient rule, e.g. d tan( z ) dz = d dz sin( z ) cos( z ) = cos( z ) cos( z ) - sin( z )( - sin( z )) cos 2 ( z ) = 1 cos 2 ( z ) = sec 2 z (No surprises there!) 9. sinh( z ), cosh( z ) ( hyperbolic sine and cosine ) Definition. cosh( z ) = e z + e - z 2 , sin( z ) = e z - e - z 2 Domain: these functions are entire. d cosh( z ) dz = sinh( z ) , d sinh( z ) dz = cosh( z ) Other key properties of cosh and sinh:
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2 ANALYTIC FUNCTIONS 14 - cosh 2 ( z ) - sinh 2 ( z ) = 1 - For real x , cosh( x ) is real and positive, sinh( x ) is real. - cosh( iz ) = cos( z ), sinh( z ) = - i sin( iz ). 10. log( z ) (See Topic 1.) Definition. log( z ) = log( | z | ) + i arg( z ). Branch: Any branch of arg( z ). Domain: C minus a branch cut where the chosen branch of arg( z ) is discontinuous. d dz log( z ) = 1 z 11. z a (any complex a ) (See Topic 1.) Definition. z a = e a log( z ) . Branch: Any branch of log( z ). Domain: Generally the domain is C minus a branch cut of log. If a is an integer 0 then z a is entire. If a is a negative integer then z a is defined and analytic on C = { 0 } .
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