Complex Analysis

Math 312 Spring 1998  MWF 10:30am - 11:25am
Buckmire  Fowler 112

Class 12 (Monday February 9)

SUMMARY The Complex Exponential and other elementary functions CURRENT READING Brown & Curchill, pages 65-75 NEXT READING Brown & Curchill pages 75-82

Now that we know something about analytic functions in general we need to expand our repertoire of complex functions.

The Complex Exponential e^{z} The complex version of the exponential function is defined like this: e^{z} = e^{x+iy} = e^{x}(cos y + i sin y), where |e^{z}| = e^{x} and arg(e^{z}) = y + 2kπ, k = 0, ±1, ±2, … arg(e^{z}) = y + 2kπ (k = 0, ±1, ±2, …) \underline{Exercise} Show that f(z) = e^{z} is an entire function and that f′(z) = e^{z} Take some time ( 3 minutes) to try and prove this. You will have to answer the questions:

1: What is an entire function?

2: How do you show that a function is analytic?

3: Do the real and complex parts of e^{z} obey the CRE?

More Properties of e^{z}

• e^{z} is never zero

• e^{z} = 1 ⟺ z = 2πki

• e^{z_{1}} = e^{z_{2}} ⟺ z_{1} = z_{2} + 2kπi, where k ∈ Z

• e^{z} is a periodic function with period 2πi

Sketch a fundamental region for e^{z} below Other elementary functions

Once we have a handle on exp z we can use it to define other functions, most notably sin z and cos z

sin z = \frac{e^{iz} − e^{−iz}}{2i} , cos z = \frac{e^{iz} + e^{−iz}}{2}

\underline{Groupwork} Show that \frac{d}{dz} cos z = − sin z by using the definition of cos z

There a whole bunch of typical trigonometric identities which are valid for complex trig functions. Most of these can be proved using the definitions involving exponentials. For example, tanz, sec z are analytic everywhere except at the zeroes of cos z. \underline{Groupwork} Find the zeroes of cos z and sin z

The usual rules of derivatives of the trig functions remain valid for their complex counterparts. Complex Trigonometric Identities

sin(z + 2π) = sin z, cos(z + 2π) = cos z
sin(−z) = − sin z, cos(−z) = cos z
sin^{2} z + cos^{2} z = 1, tan^{2} z + 1 = sec^{2} z
sin 2z = 2 sin z cos z, cos 2z = cos^{2} − sin^{2} z
sec z = \frac{1}{cos z} , tan z = \frac{sin z}{cos z}
\frac{d}{dz} tan z = sec^{2}z, \frac{d}{dz} sec z = secz tan z \frac{d}{dz} sin z = cos z, \frac{d}{dz} sin z = cos z

Similarly the hyperbolic trigonometric functions can be defined using the complex exponential and the newly-defined complex trig functions

sinh z = \frac{e^{z} − e^{−z}}{2} , cosh z = \frac{e^{z} + e^{−z}}{2}

Complex Hyperbolic Trigonometric Identities

sinh z = −i sin iz, cosh z = cos(iz)
\frac{d}{dz} sinh z = cosh z, \frac{d}{dz} cosh z = sinh z