Complex exponential Complex exponential Trigonometric and hyperbolic functions Trigonometric and hyperbolic functions Definition Complex logarithm Complex logarithm Properties Complex power function Complex power function 1. Complex exponential

The exponential of a complex number z = x + iy is defined as

Chapter 13: Complex Numbers exp(z)=exp(x + iy)=exp(x) exp(iy) =exp(x)(cos(y)+i sin(y)) . Sections 13.5, 13.6 & 13.7

As for real numbers, the exponential function is equal to its derivative, i.e. d exp(z)=exp(z). (1) dz

The exponential is therefore entire.

You may also use the notation exp(z)=ez .

Chapter 13: Complex Numbers Chapter 13: Complex Numbers

Complex exponential Complex exponential Trigonometric and hyperbolic functions Definition Trigonometric and hyperbolic functions Trigonometric functions Complex logarithm Properties Complex logarithm Hyperbolic functions Complex power function Complex power function Properties of the exponential function 2. Trigonometric functions

The exponential function is periodic with period 2πi: indeed, for any integer k ∈ Z, The complex sine and cosine functions are defined in a way similar to their real counterparts, exp(z +2kπi)=exp(x)(cos(y +2kπ)+i sin(y +2kπ)) eiz + e−iz eiz − e−iz =exp(x)(cos(y)+i sin(y)) = exp(z). cos(z)= , sin(z)= . (2) 2 2i Moreover,
  The tangent, cotangent, secant and cosecant are defined as | exp(z)| = | exp(x)||exp(iy)| = exp(x) cos2(y)+sin2(y) usual. For instance, x e z . = exp( )=exp( ( )) sin(z) 1 tan(z)= , sec(z)= , etc. As with real numbers, cos(z) cos(z)

exp(z1 + z2)=exp(z1) exp(z2); exp(z) =0.

Chapter 13: Complex Numbers Chapter 13: Complex Numbers Complex exponential Complex exponential Trigonometric and hyperbolic functions Trigonometric functions Trigonometric and hyperbolic functions Trigonometric functions Complex logarithm Hyperbolic functions Complex logarithm Hyperbolic functions Complex power function Complex power function Trigonometric functions (continued) 3. Hyperbolic functions

The complex hyperbolic sine and cosine are defined in a way The rules of differentiation that you are familiar with still similar to their real counterparts, work. ez + e−z ez − e−z cosh(z)= , sinh(z)= . (3) Example: 2 2 Use the definitions of cos(z) and sin(z), The hyperbolic sine and cosine, as well as the sine and cosine, eiz + e−iz eiz − e−iz cos(z)= , sin(z)= . are entire. 2 2i We have the following relations to find (cos(z)) and (sin(z)). cosh(iz)=cos(z), sinh(iz)=i sin(z), θ Show that Euler’s formula also works if is complex. (4) cos(iz)=cosh(z), sin(iz)=i sinh(z).

Chapter 13: Complex Numbers Chapter 13: Complex Numbers

Complex exponential Complex exponential Trigonometric and hyperbolic functions Definition Trigonometric and hyperbolic functions Definition Complex logarithm Principal value of ln(|z|) Complex logarithm Principal value of ln(|z|) Complex power function Complex power function 4. Complex logarithm Principal value of ln(z)

The logarithm w of z = 0 is defined as We define the principal value of ln(z), Ln(z), as the value of ew = z. ln(z) obtained with the principal value of arg(z), i.e. Since the exponential is 2πi-periodic, the complex logarithm is multi-valued. Ln(z) = ln(|z|)+i Arg(z). iθ Solving the above equation for w = wr + iwi and z = re y gives Note that Ln(z)  Arg(z)=π ewr = r jumps by −2πi when 1 ew = ewr eiwi = reiθ =⇒ , wi = θ +2pπ one crosses the 0 1 x negative real axis Arg(z)–> - π which implies wr =ln(r)andwi = θ +2pπ, p ∈ Z. from above. Therefore, ln(z) = ln(|z|)+i arg(z). The negative real axis is called a branch cut of Ln(z).

Chapter 13: Complex Numbers Chapter 13: Complex Numbers Complex exponential Complex exponential Trigonometric and hyperbolic functions Definition Trigonometric and hyperbolic functions Definition Complex logarithm Principal value of ln(|z|) Complex logarithm Principal value of ln(|z|) Complex power function Complex power function Principal value of ln(z) (continued) Properties of the logarithm

You have to be careful when you use identities like Recall that   z |z| i z . z Ln( ) = ln( )+ Arg( ) ln(z z ) = ln(z )+ln(z ), or ln 1 =ln(z )−ln(z ). 1 2 1 2 z 1 2 Since Arg(z)=arg(z)+2pπ, p ∈ Z, we therefore see that 2 ln(z) is related to Ln(z)by They are only true up to multiples of 2πi.

ln(z) = Ln(z)+i 2pπ, p ∈ Z. For instance, if z1 = i = exp(iπ/2) and z2 = −1=exp(iπ), Examples: π ln(z1)=i +2p1iπ, ln(z2)=iπ +2p2iπ, p1, p2 ∈ Z, Ln(2) = ln(2), but ln(2) = ln(2) + i 2pπ, p ∈ Z. 2 Find Ln(−4) and ln(−4). and 3π ln(z z )=i +2p iπ, p ∈ Z, Find ln(10 i). 1 2 2 3 3

but p3 is not necessarily equal to p1 + p2.

Chapter 13: Complex Numbers Chapter 13: Complex Numbers

Complex exponential Complex exponential Trigonometric and hyperbolic functions Definition Trigonometric and hyperbolic functions Definition Complex logarithm Principal value of ln(|z|) Complex logarithm Complex power function Complex power function Properties of the logarithm (continued) 5. Complex power function

Moreover, with z1 = i = exp(iπ/2) and z2 = −1=exp(iπ), π If z =0and c are complex numbers, we define Ln(z )=i , Ln(z )=i π, 1 2 2 zc =exp(c ln(z)) and π Ln(z z )=−i =Ln( z )+Ln(z ). =exp(c Ln(z)+2pcπi) , p ∈ Z. 1 2 2 1 2

However, every branch of the logarithm (i.e. each expression For c ∈ C, this is again a multi-valued function, and we define of ln(z) with a given value of p ∈ Z) is analytic except at the the principal value of zc as branch point z =0andonthebranchcutofln(z). In the c domain of analyticity of ln(z), z = exp (c Ln(z)) d 1 (ln(z)) = . (5) dz z

Chapter 13: Complex Numbers Chapter 13: Complex Numbers