tma4115 - calculus 3 · tma4115 - calculus 3 lecture 3, jan 23 toke meier carlsen norwegian...

TMA4115 - Calculus 3Lecture 3, Jan 23

Toke Meier CarlsenNorwegian University of Science and TechnologySpring 2013

www.ntnu.no TMA4115 - Calculus 3, Lecture 3, Jan 23

Review of last week

Last week weintroduced complex numbers, both in a geometric wayand in an algebraic way,defined Re(z), Im(z), |z| and arg(z) for a complexnumber z,defined addition and multiplication of complex numbers,defined complex conjugation,introduced polar representation of complex numbers,computed powers of complex numbers,defined and computed roots of complex numbers.

www.ntnu.no TMA4115 - Calculus 3, Lecture 3, Jan 23, page 2

Today’s lecture

Today we shalluse complex numbers to solve polynomial equations,look at the fundamental theorem of algebra,introduce the complex exponential function,and study extensions of trigonometric functions to thecomplex numbers.


Solutions to second degreeequations

If a,b, c are complex numbers and a 6= 0, then the solutions

to the equation az2 + bz + c = 0 are z =−b ±

√b2 − 4ac

2a.


Problem 1 from the exam fromJune 2012

Solve w2 = (−1 + i√

3)/2. Find all solutions of the equationz4 + z2 + 1 = 0 and draw them in the complex plane. Writethe solutions in the form x + iy .


Roots of polynomials

If P(z) is a polynomial, then a solution to the equationP(z) = 0 is called a root (or zero) of P(z).z0 is a root of P(z) if and only if (z − z0) is a factor of P(z)(i.e., if P(z) = (z − z0)Q(z) for some polynomial Q(z)).


The fundamental theorem ofalgebra

Every complex polynomial of degree 1 or higher has a leastone complex root.


Roots of real polynomials

If w is a root of a real polynomial∑n

k=0 akzk , then w is also aroot of

∑nk=0 akzk .


Exercise 34 on page xxvii

Check that z1 = 1−√

3i is a zero ofP(z) = z4 − 4z3 + 12z2 − 16z + 16, and find all the zeros ofP.


Complex functions

A complex function f is a rule that assigns a unique complexnumber f (z) to each number z in some set of complexnumbers (called the domain of f ).

Examples of complex functionsf (z) = Re(z)g(z) = Im(z)h(z) = |z|j(z) = Arg(z)k(z) = zp(z) = z2 − 4z + 6


Graphic representations ofcomplex functions

We cannot draw the graph of a complex function sincewe would need 4 dimensions to do that.Instead, we can graphically represent the behavior of acomplex function w = f (z) by drawing the z-plane andthe w-plane separately, and showing the image in thew-plane of certain, appropriately chosen set of points inthe z-plane.


Complex functions

Limits, continuity and differentiability of complex functionscan be defined just as for real functions.

Examples of complex functionsEvery complex polynomial is differentiable, and hencecontinuous.The functions f (z) = Re(z), g(z) = Im(z), h(z) = |z| andk(z) = z are continuous, but not differentiable.The function j(z) = Arg(z) is not continuous.


The exponential function

One can show that the series∞∑

n=0

zn

n!converges absolutely for

all complex numbers z.

We denote the sum of∞∑

n=0

zn

n!as the exponential function ez .

ez is also the limit limn→∞

(1 +

zn

)n.


The exponential function and cosand sin

If y is a real number, then

eiy =∞∑

n=0

(iy)n

n!=∞∑

n=0

(iy)2n

(2n)!+∞∑

n=0

(iy)2n+1

(2n + 1)!=

∞∑n=0

(−1)ny2n

(2n)!+ i

∞∑n=0

(−1)ny2n+1

(2n + 1)!= cos(y) + i sin(y).


The exponential function

One can show that ez1+z2 = ez1ez2. It follows thatez = exeiy = ex(cos y + i sin y) for z = x + iy .


The exponential function and polarrepresentation

If z 6= 0, then

z = |z|(cos(arg(z)) + i sin(arg(z)))

=eln(|z|)ei arg(z) = eln(|z|)+i arg(z).

The exponential function is not injective (becauseex+iy = ex+i(y+2π)), and does therefore not have an inverse.


Properties of the exponentialfunction

If z = x + iy , thenez = ez

Re(ez) = ex cos yIm(ez) = ex sin y|ez | = ex

arg(ez) = yOne can also show that ez is differentiable and thatddz ez = ez .


Sine and cosine


n=0

(−1)nz2n

(2n)!and

∞∑n=0

(−1)nz2n+1

(2n + 1)!converge absolutely for all complex numbers

z.


n=0

(−1)nz2n

(2n)!as cos(z), and the sum

of∞∑

n=0

(−1)nz2n+1

(2n + 1)!as sin(z).


Properties of sin and cos

If z is a complex number, then

cos z =eiz + e−iz

2and sin z =

eiz − e−iz

2i.

sin and cos are periodic with period 2π.sin and cos are differentiable and d

dz sin z = cos z andddz cos z = − sin z.


Hyperbolic sine and cosine


n=0

z2n

(2n)!and

∞∑n=0

z2n+1

(2n + 1)!converge absolutely for all complex numbers z.


n=0

z2n

(2n)!as cosh(z), and the sum of

∞∑n=0

z2n+1

(2n + 1)!as sinh(z).


Properties of sinh and cosh

If z is a complex number, then

cosh z =ez + e−z

2and sinh z =

ez − e−z

2.

sinh and cosh are periodic with period 2πi .sinh and cosh are differentiable and d

dz sinh z = cosh zand d

dz cosh z = sinh z.


Plan for tomorrow

Tomorrow we shallstudy second-order linear differential equations,introduce the Wronskian,completely solve second-order homogeneous lineardifferential equations with constant coefficients.

Section 4.1 and 4.3 in “Second-Order Equations” (pagesxxxv-xlv and xlix-lv).


Second-order linear differentialequations

A second-order linear differential equation is a differentialequation with can be written on the form

y ′′ + p(t)y ′ + q(t)y = g(t).

Such an equation is homogeneous if g(t) = 0.


tma4115 - calculus 3 · tma4115 - calculus 3 lecture 3, jan 23 toke meier carlsen norwegian...

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