MA 201 Complex Analysis Lecture 6: Elementary functions

Lecture 6 Elementary functions The Exponential Function

Recall: iy Euler’s Formula: For y ∈ R, e = cos y + i sin y x+y x y and for any x, y ∈ R, e = e e .

Definition: If z = x + iy, then ez or exp(z) is defined by the formula

ez = e(x+iy) = ex (cos y + i sin y).

Image of following sets under exponential function:

x {(x, y0): x ∈ R} 7−→ {(r, θ): r = e , x ∈ R, θ = y0}.

x0 {(x0, y): y ∈ R} 7−→ {(r, θ): r = e , θ ∈ R}.

{(x, y): a ≤ x ≤ b, c ≤ y ≤ d} 7−→ {(r, θ): ea ≤ r ≤ eb, c ≤ θ ≤ d}.

Lecture 6 Elementary functions Properties of Exponential Function

e0 = 1

ez = ez¯ , |ez | ≤ e|z|.

z z x iy x e 6= 0, for all z ∈ C. Look at |e | = |e ||e | = e 6= 0.

z+w z w For all z, w ∈ C, e = e e .

Indeed, if z = x + iy, w = s + it then,

ez+w = e(x+s)+i(y+t) = e(x+s)[cos(y + t) + i sin(y + t)] = ex es [(cos y cos t − sin y sin t) + i(sin y cos t + cos y sin t)] = [ex (cos y + i sin y)][es (cos t + i sin t)] = ez ew .

Lecture 6 Elementary functions Properties of Exponential function

A function f : C → C is called periodic if there is a w ∈ C (called a period) such that f (z + w) = f (z) for all z ∈ C.

ez is periodic function with period 2πi.

ez is not injective unlike real exponential.

z x x Since e = e cos y + ie sin y satisfies C-R equation on C and has continuous first order partial derivatives. Therefore ez is an entire function.

d z ∂ x ∂ x z In fact d z e = ∂x (e cos y) + i ∂x (e sin y) = e .

Lecture 6 Elementary functions Trigonometric Functions

Define 1 1 sin z = (eiz − e−iz ); cos z = (eiz + e−iz ). 2i 2 Properties: sin2 z + cos2 z = 1

sin(z + w) = sin z cos w + cos z sin w

cos(z + w) = cos z cos w − sin z sin w

sin(−z) = − sin z and cos(−z) = cos z

sin(z + 2kπ) = sin z, and cos(z + 2kπ) = cos z

1 sin z = 0 ⇐⇒ z = nπ and cos z = 0 ⇐⇒ z = (n + 2 )π, , n ∈ Z.

Lecture 6 Elementary functions Trigonometric functions

sin z, cos z are entire functions.

d d dz (sin z) = cos z and dz (cos z) = − sin z.

sin z and cos z are unbounded functions.

Define:

sin z tan z = cos z cos z cot z = sin z

1 sec z = cos z

1 csc z = . sin z

Lecture 6 Elementary functions Trigonometric functions

Hyperbolic Trigonometric functions: Define

ez − e−z ez + e−z sinh z = ; and cosh z = . 2 2

Properties: sinh z, cosh z are entire functions. cosh2 z − sinh2 z = 1. sinh(−z) = − sinh z, cosh(−z) = cosh z, sinh(z + 2kπ) = sinh z, cosh(z + 2kπ) = cosh z, k ∈ Z. sinh(iz) = i sin z and cos(iz) = cos z sin z = sin(x + iy) = sin x cosh y + i cos x sinh y, cos z = cos(x + iy) = cos x cosh y − i sin x sinh y.

Lecture 6 Elementary functions Complex Logarithm

z ∗ Note that e is an onto function from C to C = C \{0}. In fact if ∗ iθ w ∈ C then w = |w|e where θ ∈ (−π, π]. If we set z = log |w| + iθ then, ez = elog |w|+iθ = elog |w|eiθ = w.

z z+2πik z e is not an injective function as e = e , k ∈ Z. But if we restrict the domain then it becomes injective.

In particular if H = {z = x + iy : −π < y ≤ π} z then z → e is a bijective function from H to C \{0}.

Question: What is the inverse of this function?

Lecture 6 Elementary functions Complex Logarithm

∗ Definition: For z ∈ C , define log z = ln |z| + i arg z.

ln |z| stands for the real logarithm of |z|.

Since arg z = Arg z + 2kπ, k ∈ Z it follows that log z is not well defined as a function. (multivalued)

∗ For z ∈ C , the principal value of the logarithm is defined as Log z = ln |z| + i Argz.

∗ Log : C → {z = x + iy : −π < y ≤ π} is well defined (single valued).

Log z + 2kπi = log z for some k ∈ Z.

Lecture 6 Elementary functions Complex Logarithm

If z 6= 0 then eLog z = eln |z|+i Argz = z

What about Log (ez )?.

Suppose x is a positive real number then Log x = ln x + i Argx = ln x.

π iπ Log i = ln |i| + i 2 = 2 , Log (−1) = ln | − 1| + iπ = iπ,

−π iπ Log (−i) = ln | − i| + i 2 = − 2 , Log (−e) = 1 + iπ

Lecture 6 Elementary functions Complex Logarithm

The function Log z is not continuous on the negative real axis − R = {z = x + iy : x < 0, y = 0}. To see this consider the point z = −α, α > 0. Consider the sequences

i(π− 1 ) i(−π+ 1 ) {an = αe n } and {bn = αe n }.

Then lim an = z = lim bn n→∞ n→∞ but 1 lim Log an = lim ln α + i(π − ) = ln α + iπ n→∞ n→∞ n and lim Log bn = ln α − iπ. n→∞

Lecture 6 Elementary functions Complex Logarithm

∗ − iθ The function z → Log z is analytic on the set C \ R . Let z = re 6= 0 and θ ∈ (−π, π). Then

Log z = ln r + iθ = u(r, θ) + iv(r, θ)

with u(r, θ) = ln r and v(r, θ) = θ. Then 1 1 1 u = v = and v = − u . r r θ r r r θ

The identity Log (z1z2) = Logz1 + Log z2 is not always valid. However, the above identity is true if and only if Arg z1 + Arg z2 ∈ (−π, π] (why?).

Lecture 6 Elementary functions Complex Logarithm

Branch of a multiple valued function: Let F be a multiple valued function defined on a domain D. A function f is said to be a branch of the multiple valued function F if in a domainD0 ⊂ D if f (z) is single valued and analytic in D0.

Branch Cut: The portion of a line or a curve introduced in order to define a branch of a multiple valued function is called branch cut.

Branch Point: Any point that is common to all branch cuts is called a branch point.

Lecture 6 Elementary functions Complex Exponents

Let w ∈ C. For any z 6= 0, define z w = exp(w log z),

where “exp” is the exponential function and log is the multiple valued logarithmic function.

z w is a multiple valued function.

i π π i = exp[i log i] = exp[i(log 1 + i 2 )] = exp(− 2 ).

Lecture 6 Elementary functions