Maths 340: Calculus

(x₁, y₁)÷(x₂, y₂)

((x₁x₂+y₁y₂)/(x₂²+y₂²), (y₁x₂-x₁y₂)/(x₂²+y₂²)

The Binomial Theorem

(a+b)ⁿ=∑(k=0 to n) (n choose k)a^(n-k)b^k

Imaginary part of x + iy

Im(z)=y

The principle value of ln(z)

Ln(z)=ln|z| + i Arg(z)

||a×b||

The area of the parallelogram the vectors a and b span

Euler's number, e

e=lim(n→∞)(1+1/n)ⁿ

z₁z₂ in Euler form

r₁r₂e^(i(θ₁+θ₂))

The definition of tan(z) extended to complex arguments

sin(z)/cos(z)

The derivative of u.v where u and v are functions of T

u'.v+u.v'

The derivative of u.v×w where u, v and w are functions of T

u'.v×w+u.v'×w+u.v×w'

zⁿ in Euler form for n∈Z

zⁿ=rⁿe^(inθ)

Multiplication of complex numbers in Euler form

z₁z₂=(r₁e^(iθ₁))(r₂e^(iθ₂)=r₁r₂e^i(θ₁+θ₂)

Modulus of z = x+iy

|z|=√(x²+y²)

The triangle inequality

|z₁+z₂|≤|z₁|+|z₂| for z₁, z₂∈C

i

√(-1)

Parameterise a curve with respect to t

(1-t)×(start point) + t×(end point)

THEOREM: Let z=x+iy and f(z)=u(x,y)+iv(x,y). In order for the first partial derivatives of u and v to exist and satisfy the Cauchy Riemann equations at z, the following conditions must hold:

(a) f is defined and continuous in an open disk B(z, r) for some r>0 and (b) f is differentiable at z

de Moivre's formula

(cosθ+isinθ)ⁿ=cos(nθ)+isin(nθ)

The definition of cosh(z) extended to complex arguments

(e^z+e^(-z))/2

The definition of sinh(z) extended to complex arguments

(e^z-e^(-z))/2

(x₁, y₁) + (x₂, y₂)

(x₁+x₂, y₁+y₂)

(x₁, y₁) - (x₂, y₂)

(x₁-x₂, y₁-y₂)

(x₁, y₁)×(x₂, y₂)

(x₁x₂ - y₁y₂, x₁y₂ + y₁x₂)

1/i

-i

If we reverse the orientation of C, and we call this reverse path -c, then ∫(-c) f(z)dz=

-∫(c) f(z) dz

An important little integral: ∫(closed curve)z^m dz

0 for m∈Z\{-1} and 2πi for m=-1

d/dx ln(x)

1/x

A domain

A domain is an open, connected set

A continuous function

A function f:D→C is continuous at z₀∈D if for every ε>0 there exists a δ>0 such that whenever z∈B(z₀,δ)∩D we have |f(z)-f(z₀)|<ε

Norm of a vector

A norm is a function that assigns a strictly positive length or size to each vector in the vector space. We use the Euclidean norm. For a vector v=[a, b, c, d], ||v||=√(a²+b²+c²+d²)

An interior point

A point z₀∈D⊆C is called an interior point of D if there exists an open r-disk centred at z₀ that lies entirely in D. That is, B(z₀, r)⊆D.

A connected subset

A subset D⊆C is called connected if any 2 points in D can be joined by a finite number of line segments which lie entirely in D.

An open subset

A subset, D, of C is called open if every point of D is an interior point of D

Fundamental Theorem of Algebra

Every polynomial equation a₀+a₁z+a₂z²+...+anzⁿ=0 (a∈C) of positive degree has a (complex) root

Taylor's Formula

Gives an approximation of a function. f(x)≈f(a)+(x-a)f'(a)+(x-a)²/2!*f''(a)+...+(x-a)ⁿ⁻¹/(n-1)!*fⁿ⁻¹(a). We denote this approximation by P(n-1)(x) and refer to it as the Taylor polynomial of degree n-1

Harmonic functions

Harmonic functions are the solutions of Laplace's equation which have continuous second partial deriatives

Principal argument of z, Arg(z)

If 0≤θ≤2π, then θ is called the principal argument of z. The principal argument of θ is uniquely determined by sinθ=y/|z| and cosθ=x/|z|

An analytic function

If f is differentiable at every point in D then it is said to be analytic in D

Liouville's Theorem

If f:C→C is analytic and bounded, then it is a constant.

Harmonic conjugate

If two harmonic functions u and v satisfy the Cauchy-Riemann equations in a domain D, then v is a harmonic conjugate of u in D. If v is a harmonic conjugate of u, then v is harmonic and the Cauchy-Riemann equations are satisfied. Moreover, f(z)=u+iv

Linear independence

If u and v are vectors, u and v are linearly independent if the only solution to 0=xu+yv is x=y=0

Arg(z)

Is the principle value of arg(z). It is the unique value of θ satisfying 0≤θ≤2π

Generalized Cauchy's Integral Formula

Let D be a simply connected domain and a∈D. If f:D→C is analytic, then f^(n)(a)=n!/(2πi)∫(f(z)/(z-a)^(n+1) dz for every n∈N₀ and every closed contour C in D enclosing a.

Fundamental Theorem of Complex Integral Calculus

Let D be a simply connected domain f:D→C analytic, z₀∈D. Then (1) G(z)=∫(from z₀ to z) f(w)dw is analytic on D and G'(z)=f(z) and (2) If F(z) is any anti-derivative of f (that is, F'=f) then ∫(from z₀ to z)=F(z)-F(z₀)

Differentiability of a complex function

Let D⊆C be a domain and let f:D→C be a complex function. Then f is said to be differentiable at z∈D if [lim(∆z→0) (f(z+∆z)-f(z))/∆z] exists. In this case it is denoted f'(z) and is called the derivative of f at z.

Geometric interpretation of dot product

Let a and b be vectors and θ is the angle between them. Then a.b=||a|| ||b|| cosθ

Principle of Deformation of Contours

Let f(z) be analytic in a domain bounded by two simple closed curves: C₁ and C₂ (where C₂ lies inside C₁) and on these curves. Then ∫(closed c₁) f(z)dz=∫(closed c₂) f(z)dz

Cauchy Theorem

Let f(z) be analytic with continuous derivatives f'(z) on and inside a simple closed curve C. Then ∫(closed c) f(z) dz=0

An open r-disk

Let r>0 and z₀∈C. The set of points B(z₀, r)={z∈C: |z-z₀|<r} is called an open r-disk centred at z₀

Cross product

Let u and v be n-dimensional vectors. Their cross-product, denoted by u×v is defined as ||u|| ||v||sin(θ) if u, v, θ≠0 and θ≠π where e is a unit vector normal to the plane of u and v, forming a right handed system. To calculate the cross product, we can form a matrix where the top row are vectors i, j, k and the next two rows are vectors u and v

Scalar triple product u.(v×w)

Let u, v, w be three n-vectors. u.(v×w) is the parallelopiped defined by the three vectors. It is equal to the determinant of the matrix where u, v and w are the row of the matrix

Orthogonality

Orthogonality is the relation of two lines at right angles and the generalisation of this to three dimensions. Two vectors are orthogonal if and only if their dot product is 0

Let P(z)=c₀+c₁z+c₂z²+...+cnzⁿ, c∈R be a polynomial in z∈C of degree n≥1. P(z₀) = 0, then

P(conjugate of z₀)=0. That is, the complex roots of P occur in conjugate pairs

Find the nth root of z

Put z in Euler form. z^(1/n)=|z|^(1/n)×e^(i×θ/n+i×2kπ/n)

Real part of x + iy

Re(z)=x

The Principle of Path Independence

Suppose D⊆C is a simply connected domain and f:D→C is analytic and let z₁, z₂∈D. Then for any paths C₁, C₂ in D from z₁ to z₂ we have ∫(c₁) f(z)dz=∫(c₂) f(z)dz

A complex function as a pair of real functions

Suppose that f:D→C is a function and D⊆C. If z∈D and w=f(z)∈C, write z=x+iy and w=u+iv for x, y, u, v∈R. Then f(x+iy)=u+iv=u(x,y)+iv(x, y). If we identify C with R² then each complex function f:D(⊆C)→C corresponds to pairs of real functions u, v: D(⊆R²)→R where u=Re(f) and v=Im(f)

THEOREM: Let D⊆R² be a domain and suppose u:D→R and v:D→R are continuous and have continuous first partial derivatives that satisfy the Cauchy Riemann equations on D. Then...

The complex function defined by f(x)=u(x, y)+iv(x,y), z=x+iy is analytic in D.

The complex logarithm

The complex logarithm is the multi-valued function defined by ln(z)=ln|z|+i arg(z), z≠0. If z=re^(iθ) then we have infinitely many values for ln(z), namely ln(r)+i(θ+2kπ). However, for each of these values we still have e^(ln(z)) = z

If f is analytic in a domain D, then ...

The first partial derivatives of u and v exist and satisfy the Cauchy Riemann equations at all points of D AND f is differentiable at every point of D AND hence the partial derivatives of u and v exist and satisfy the Cauchy Riemann equations at every point of D AND u and v satisfy Laplace's equation in D and have continuous second partial derivatives in D

THEOREM: Let z=x+iy and f(z)=u(x,y)+iv(x,y). Suppose that (a) f is defined and continuous in an disk B(z, r) for some r>0 and (b) f is differentiable at z. Then

The first partial derivatives of u and v exist and satisfy the Cauchy Riemann equations at z.

The remainder of the Taylor polynomial

The remainder term, Rn(x) is defined as the difference between f(x) and P(n-1)(x). Rn(x)=f^(n)(η)/n!*(x-a)^n where η∈[a, x]

THEOREM: Suppose u is harmonic on a rectangle D=(a,b)×(c,d)⊆R². Then ...

There exists a harmonic conjugate v of u. Furthermore, v is unique up to a constant. f=u+iv is analytic on D.

Equality of complex numbers

Two complex numbers are equal if and only if their real parts and imaginary parts are equal

Suppose the curve C is of length L and let M=max{|f(z)|:z∈C}. Then...

We can bound |∫(c) f(z)dz|≤ML

Dot product of vectors (Algebraic interpretation)

[a, b, c, d].[w, x, y, z] = aw+bx+cy+dz. The dot product is not defined if vectors have a different number of components.

Laplace's equation in D

\/²u=uxx+uyy=0 and \/²u=vxx+vyy=0

Argument of conjugate

arg(z⁻)=-arg(z)

The definition of cot(z) extended to complex arguments

cos(z)/sin(z)

The definition of cosine extended to complex arguments

cos(z)=(e^(iz)+e^(-iz))/2

The definition of sine extended to complex arguments

cos(z)=(e^(iz)-e^(-iz))/2i

Equation for a plane

d=ax+by+cd. By substituting in known points you can find the equation for the plane.

The derivative of u(f(T)) where u and f are functions of T

du/dT f'(T)

Let u be a vector whose components depend on T. The derivative of u with respect to T is denoted by du/dT and is defined as:

du/dT=lim(∆T→0) of (u(T+∆T)-u(T))/∆T. We differentiate each component of u with respect to T

If c∈C then we define z^c

e^(cln(z)), z≠0. The principle value of z^c is given by e^(cLn(z))

Euler Formula

e^(iθ) = cos(θ) + i sin(θ)

Periodicity of e^(iθ)

e^(θ+2kπ)i = e^(iθ) for k∈Z

The Taylor series for exp(x)

e^x=∑(from n=0 to ∞) xⁿ/n! = 1+x+x²/2+x³/3!+x⁴/4!+ ...

Analytic at a point

f is analytic at a point z∈D if it is differentiable in some open disk centred at z

The derivative of fu where f and u are functions of T

f'u+fu'

The natural logarithm

ln: R>0→R is the inverse of the exponential function. If x=e^y then y=ln(x)

Polar form of the Cauchy-Riemann equations

rU(subscript)r=V(subscript)θ and rV(subscript)r=-U(subscript)θ

Let n∈N. the n-th roos of z are all complex numbers satisfying wⁿ=z. If z=re^(iθ), then its n-th roots are

r^(1/n)e^(iθ/n+i2kπ/n), k = 0, 1, ..., n-1

x in Polar coordinates

rcos(θ)

y in Polar coordinates

rsin(θ)

z₁/z₂ in Euler form

r₁/r₂ e^(i(θ₁-θ₂))

The derivative of u×v where u and v are functions of T

u'×v+u×v'

PROPOSITION: f:D→C is continuous at z₀=x₀+iy₀ ∈ D if and only if...

u=Re(f) and v=Im(f) are continuous at (x₀, y₀)

The Cauchy Riemann Equations

ux(x,y)=vy(x,y) and vx(x,y)=-uy(x,y)

The complex conjugate of z = x+iy = re^(iθ)

x-iy = re^(-iθ)

cos(θ) in Cartesian coordinaes

x/√(x²+y²)

The quadratic formula

x=(-b±√(b²-4ac))/2a

Parameterize a circular path

x=rcosθ, y=rsinθ OR z=re^(it)

tan(θ) in Cartesian coordinates

y/x

sin(θ) in Cartesian coordinates

y/√(x²+y²)

Equation for a line between two points

y=mx+c, m=(y₂-y₁)/(x₂-x₁)

Complex conjugation in the Argand plane

z is reflected in the real axis

Parameterize a line segment from z₀ to z₁

z=(1-t)×z₀+t×z₁ with t∈[0,1]

z=x+iy in polar coordinates

z=r cosθ+i r sinθ, where r=|z|

z=x+iy in Euler form

z=r×e^(iθ)

Taking complex number to a power

z^m=(r×e^(iθ))^m=r^m×e^(iθm)

Division of complex numbers in Euler form

z₁/z₂=r₁/r₂×e^i(θ₁-θ₂)

The derivative of αu+βv where α and β are constant scalars

αu'+βv'

∫(c) αf(z) dz, α∈C

α∫(c) f(z) dz,

arg(z)

θ = argument of z = arg(z). It gives the angle which the line from the origin to z makes with the positive real axis. Because a complete rotation about 0 makes the complex number unchanged, there are many choices that we could make for θ by continually circling the origin. arg(z)=θ+2nπ, n∈Z

Argument of z, arg(z)

θ, the angle between z and the x axis. θ is not unique. θ=arctan(y/x)

∫(c) f(z)±g(z) dz

∫(c) f(z)dz ± ∫(c) g(z)dz

THEOREM: Suppose D⊆C is a simply connected domain and f:D→C is analytic. If C is a closed contour in D, then

∫(closed c) f(z)dz=0

Suppose that C=C₁+C₂, that is, the ending point of C₁ is the starting point of C₂. Hence ∫(c) f(z) dz =

∫(c₁) f(z) dz + ∫(c₂) f(z) dz

Let f be a complex-valued function defined in a domain D⊆C. Consider the integral ∫c f(z)dz, where C is a contour in D joining z₁ to z₂, Let z=x+iy so that f(z)=u(x, y)+iv(x,y) and dz=dx+idy. Then ∫c f(z)dz =

∫c (u+iv)(dx+idy)=∫c (udx-vdy)+i∫c (vdx+udy) where the integral is resolved into real and imaginary line integrals over path C in R² joining (x₁, y₁) to (x₂, y₂)