Intermediate mathematics/Functions

From Wikipedia:Function (mathematics)

In mathematics, a **function** is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function

f
(
x
# )

x
2

{\displaystyle f(x)=x^2}

 that relates each real number *x* to its square *x*2. The output of a function *f* corresponding to an input *x* is denoted by *f*(*x*) (read "*f* of *x*"). In this example, if the input is −3, then the output is 9, and we may write *f*(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write *f*(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function.

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## Euclids "common notions"

From Wikipedia:Euclidean geometry:
Things that do not differ from one another are equal to one another

Things that are equal to the same thing are also equal to one another

 b=c

If equals are added to equals, then the wholes are equal

 c=d

If equals are subtracted from equals, then the remainders are equal

 c=d

The whole is greater than the part.

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## Elementary algebra

From Wikipedia:Elementary algebra:
Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general (non-specified) numbers.

Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (additionw, subtractionw, multiplicationw, divisionw and exponentiationw). For example,

* Added terms are simplified using coefficients. For example,

x
x
x

{\displaystyle x + x + x}

 can be simplified as

3
x

{\displaystyle 3x}

 (where 3 is a numerical coefficient).

* Multiplied terms are simplified using exponents. For example,

x
×
x
×
x

{\displaystyle x \times x \times x}

 is represented as

x
3

{\displaystyle x^3}

* Like terms are added together, for example,

2

x
2

3
a
b
−

x
2

a
b

{\displaystyle 2x^2 + 3ab - x^2 + ab}

 is written as

x
2

4
a
b

{\displaystyle x^2 + 4ab}

, because the terms containing

x
2

{\displaystyle x^2}

 are added together, and, the terms containing

a
b

{\displaystyle ab}

 are added together.

* Brackets can be "multiplied out", using the distributive propertyw. For example,

x
(
2
x
3
)

{\displaystyle x (2x + 3)}

 can be written as

(
x
×
2
x
)
(
x
×
3
)

{\displaystyle (x \times 2x) + (x \times 3)}

 which can be written as

2

x
2

3
x

{\displaystyle 2x^2 + 3x}

* Expressions can be factored. For example,

6

x
5

3

x
2

{\displaystyle 6x^5 + 3x^2}

, by dividing both terms by

3

x
2

{\displaystyle 3x^2}

 can be written as

3

x
2

(
2

x
3

1
)

{\displaystyle 3x^2 (2x^3 + 1)}

For any function

f

{\displaystyle f}

, if

# a
b

{\displaystyle a = b}

 then:

* f
(
a
# )
f
(
b
)

{\displaystyle f(a) = f(b)}
* a
# c
b
c

{\displaystyle a + c = b + c}
* a
# c
b
c

{\displaystyle ac = bc}
* a
c

=

b
c

{\displaystyle a^c = b^c}

One must be careful though when squaring both sides of an equation since this can result is solutions that dont satisfy the original equation.

1
≠
−
1

{\displaystyle 1 \neq -1}

yet

1
2

=
−

1
2

{\displaystyle 1^2 = -1^2}

A function is an even functionw if f(x) = f(-x)

A function is an odd functionw if f(x) = -f(-x)

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## Trigonometry

The law of cosines reduces to the Pythagorean theoremw when gamma=90 degrees

c
2

=

a
2

b
2

−
2
a
b
cos
⁡
γ
,

{\displaystyle c^2 = a^2 + b^2 - 2ab\cos\gamma,}

The **law of sinesw** (also known as the "sine rule") for an arbitrary triangle states:

a

sin
⁡
A

=

b

sin
⁡
B

=

c

sin
⁡
C

=

a
b
c

2
Δ

,

{\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = \frac{abc}{2\Delta},}

where

Δ

{\displaystyle \Delta}

 is the area of the triangle

Area

=
# Δ

1
2

a
b
sin
⁡
C
.

{\displaystyle \mbox{Area} = \Delta = \frac{1}{2}a b\sin C.}

The **law of tangentsw**:

a
−
b

a
b

=

tan
⁡

[

1
2

(
A
−
B
)

]

tan
⁡

[

1
2

(
A
B
)

]

{\displaystyle \frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{2}(A+B)\right]}}

The parallelogram law reduces to the Pythagorean theorem when the parallelogram is a rectangle

(
A
B

)
2

(
B
C

)
2

(
C
D

)
2

(
D
A

)
2

=
(
A
C

)
2

(
B
D

)
2

{\displaystyle (AB)^2+(BC)^2+(CD)^2+(DA)^2=(AC)^2+(BD)^2\,}

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### Right triangles

A right triangle is a triangle with gamma=90 degrees.

For small values of x, sin x ≈ x. (If x is in radians).

CAH → cos = "adjacent" / "hypotenuse"
TOA → tan = "opposite" / "adjacent"
= cos A = b/c
= tan A = a/b

sin
⁡
# x

e

i
x

−

e

−
i
x

2
i

,

cos
⁡
# x

e

i
x

e

−
i
x

2

,

tan
⁡
# x

i
(

e

−
i
x

−

e

i
x

)

e

i
x

e

−
i
x

.

{\displaystyle \sin x = \frac{e^{ix} - e^{-ix}}{2i}, \qquad \cos x = \frac{e^{ix} + e^{-ix}}{2}, \qquad \tan x = \frac{i(e^{-ix} - e^{ix})}{e^{ix} + e^{-ix}}.}

(Note: the expression of tan(x) has i in the numerator, not in the denominator, because the order of the terms (and thus the sign) of the numerator is changed w.r.t. the expression of sin(x).)

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### Hyperbolic functions

See also: Hyperbolic angle
From Wikipedia:Hyperbolic function:

Hyperbolic functionsw are analogs of the ordinary trigonometric, or circular, functions.

* Hyperbolic sine:

sinh
⁡
# x

e
x

−

e

−
x

2

=

e

2
x

−
1

2

e
x

=

1
−

e

−
2
x

2

e

−
x

.

{\displaystyle \sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x} = \frac {1 - e^{-2x}} {2e^{-x}}.}

* Hyperbolic cosine:

cosh
⁡
# x

e
x

e

−
x

2

=

e

2
x

1

2

e
x

=

1

e

−
2
x

2

e

−
x

.

{\displaystyle \cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x} = \frac {1 + e^{-2x}} {2e^{-x}}.}

* Hyperbolic tangent:

tanh
⁡
# x

sinh
⁡
x

cosh
⁡
x

=

e
x

−

e

−
x

e
x

e

−
x

=

{\displaystyle \tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = }

=

e

2
x

−
1

e

2
x

1

=

1
−

e

−
2
x

1

e

−
2
x

.

{\displaystyle = \frac{e^{2x} - 1} {e^{2x} + 1} = \frac{1 - e^{-2x}} {1 + e^{-2x}}.}

* Hyperbolic cotangent:

coth
⁡
# x

cosh
⁡
x

sinh
⁡
x

=

e
x

e

−
x

e
x

−

e

−
x

=

{\displaystyle \coth x = \frac{\cosh x}{\sinh x} = \frac {e^x + e^{-x}} {e^x - e^{-x}} = }

=

e

2
x

1

e

2
x

−
1

=

1

e

−
2
x

1
−

e

−
2
x

,

x
≠

{\displaystyle = \frac{e^{2x} + 1} {e^{2x} - 1} = \frac{1 + e^{-2x}} {1 - e^{-2x}}, \qquad x \neq 0.}

* Hyperbolic secant:

sech
⁡
# x

1

cosh
⁡
x

=

2

e
x

e

−
x

=

{\displaystyle \operatorname{sech} x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}} = }

=

2

e
x

e

2
x

1

=

2

e

−
x

1

e

−
2
x

.

{\displaystyle = \frac{2e^x} {e^{2x} + 1} = \frac{2e^{-x}} {1 + e^{-2x}}.}

* Hyperbolic cosecant:

csch
⁡
# x

1

sinh
⁡
x

=

2

e
x

−

e

−
x

=

{\displaystyle \operatorname{csch} x = \frac{1}{\sinh x} = \frac {2} {e^x - e^{-x}} = }

=

2

e
x

e

2
x

−
1

=

2

e

−
x

1
−

e

−
2
x

,

x
≠

{\displaystyle = \frac{2e^x} {e^{2x} - 1} = \frac{2e^{-x}} {1 - e^{-2x}}, \qquad x \neq 0.}

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## Areas and volumes

The length of the circumference *C* of a circle is related to the radius *r* and diameter *d* by:

C
i
r
c
u
m
f
e
r
e
n
c
e

=
τ
# r
2
π
# r
π
d

{\displaystyle \mathrm{Circumference} = \tau r = 2 \pi r = \pi d}

where

π

{\displaystyle \pi }

 = 3.141592654

τ

{\displaystyle \tau}

 = 2 \* π
The area of a circle is:

A
r
e
a

=
π

r
2

{\displaystyle \mathrm{Area} = \pi r^2}

The surface area of a sphere is

Surface area

=
4
⋅
π

r
2

{\displaystyle \mathrm{\text{Surface area}} = 4 \cdot \pi r^2}

The surface area of a sphere 1 unit in radius is:

4
π
(
1
 unit

)
2

=
12.56637

 unit
2

{\displaystyle 4 \pi (1 \text{ unit})^2 = 12.56637 \text{ unit}^2}

The surface area of a sphere 128 units in radius is:

4
π
(
128
 unit

)
2

=
205
,
887

 unit
2

{\displaystyle 4 \pi (128 \text{ unit})^2 = 205,887 \text{ unit}^2}

The volume inside a sphere is

V
o
l
u
m
e

=

4
3

⋅
π

r
3

{\displaystyle \mathrm{Volume} = \frac{4}{3} \cdot \pi r^3}

The volume of a sphere 1 unit in radius is:

# V

4
3

⋅
π
(
1
 unit

)
3

=
4.1888

 unit
3

{\displaystyle V = \frac{4}{3} \cdot \pi (1 \text{ unit})^3 = 4.1888 \text{ unit}^3}

The moment of inertia of a hollow sphere is:

# I

2
3

m

r
2

{\displaystyle I = \frac{2}{3} m r^2\,\!}

Moment of inertia of a sphere is:

# I

2
5

m

r
2

{\displaystyle I = \frac{2}{5} m r^2\,\!}

The area of a hexagon is:

A
r
e
# a

3

3

2

a
2

=
2.59807621135
⋅

a
2

{\displaystyle Area = \frac{3 \sqrt{3}}{2} a^2 = 2.59807621135 \cdot a^2}

where *a* is the length of any side.
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### Fractals

A square that is twice as big is four times as massive because it is 2 dimensional (22 = 4).
A cube that is twice as big is eight times as massive because it is 3 dimensional (23 = 8).

A triangle that is twice as big is four times as massive.

But a Sierpiński triangle that is twice as big is exactly three times as massive. It therefore has a Hausdorff dimension of 1.5849. (21.5849 = 3)

A pyramid that is twice as big is eight times as massive.

But a Sierpiński pyramid that is twice as big is exactly five times as massive. It therefore has a Hausdorff dimension of 2.3219 (22.3219 = 5)

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## Polynomials

See also: Runge's phenomenon, Polynomial ring, System of polynomial equations, Rational root theorem, Descartes' rule of signs, and Complex conjugate root theorem
From Wikipedia:Polynomial:
A polynomialw can always be written in the form

Z
(
x
# )

a
0

a
1

x

a
2

x
2

⋯

a

n
−
1

x

n
−
1

a
n

x
n

{\displaystyle Z(x) = a\_0 + a\_1 x + a\_2 x^2 + \dotsb + a\_{n-1}x^{n-1} + a\_n x^n}

where

a
0

,
…
,

a
n

{\displaystyle a\_0, \ldots, a\_n}

 are constants called coefficients and *n* is the degreew of the polynomial.
A linear polynomial is a polynomial of degree one.
Each individual term is the product of the coefficient and a variable raised to a nonnegative integer power.
A monomial has only one term.
A binomial has 2 terms.
Fundamental theorem of algebra:

Every single-variable, degree n polynomial with complex coefficients has exactly n complex rootsw.
However, some or even all of the roots might be the same number.
A root (or zero) of a function is a value of x for which Z(x)=0.

Z
(
x
# )

a
n

(
x
−

z
1

)
(
x
−

z
2

)
⋯
(
x
−

z
n

)

{\displaystyle Z(x) = a\_n(x - z\_1)(x - z\_2)\dotsb(x - z\_n)}

If

Z
(
x
# )
(
x
−

z
1

)
(
x
−

z
2

)
k

{\displaystyle Z(x) = (x - z\_1)(x - z\_2)^k}

 then z2 is a root of multiplicity k. z2 is a root of multiplicity k-1 of the derivative (Derivative is defined below) of Z(x).
If k=1 then z2 is a simple root.
The graph is tangent to the x axis at the multiple roots of f and not tangent at the simple roots.
The graph crosses the x-axis at roots of odd multiplicity and bounces off (not goes through) the x-axis at roots of even multiplicity.
Near x=z2 the graph has the same general shape as

A
(
x
−

z
2

)
k

{\displaystyle A(x - z\_2)^k}

The complex conjugate root theorem states that if the coefficients of a polynomial are real, then the non-real roots appear in pairs of the form (*a* + *ib*, *a* – *ib*).
The roots of the formula

a

x

2

b
x
# c
0

{\displaystyle ax^{2} + bx + c = 0}

 are given by the Quadratic formulaw:

# x

−
b
±

b
2

−
4
a
c

2
a

.

{\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.}

b
2

−
4
a
c

{\displaystyle b^2 - 4ac}

 is called the discriminant.

a

x
2

b
x
c

=

a
(
x

b

2
a

)
2

c
−

b
2

4
a

=

a
(
x
−
h

)
2

k

{\displaystyle ax^2+bx+c \quad = \quad a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a} \quad = \quad a(x-h)^2+k}

This is a parabola shifted to the right h units, stretched by a factor of a, and moved upward k units.
k is the value at x=h and is either the maximum or the minimum value.
The roots of

a
0

a
1

x
⋯

a
n

x
n

,

{\displaystyle
a\_0 + a\_1 x + \cdots + a\_n x^n,
}

are the multiplicative inverses of

a
n

a

n
−
1

x
⋯

a
0

x
n

,

{\displaystyle
a\_n + a\_{n-1} x + \cdots + a\_0 x^n,
}

There is no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc). See Galois theory.
Binomial therorem:

(
x
y

)
n

=

(

n
0

)

x
n

y
0

(

n
1

)

x

n
−
1

y
1

(

n
2

)

x

n
−
2

y
2

⋯

(

n

n
−
1

)

x
1

y

n
−
1

(

n
n

)

x
0

y
n

,

{\displaystyle (x+y)^n = {n \choose 0}x^n y^0 + {n \choose 1}x^{n-1}y^1 + {n \choose 2}x^{n-2}y^2 + \cdots + {n \choose n-1}x^1 y^{n-1} + {n \choose n}x^0 y^n,

}

Where

(

n
k

)

=

n
!

k
!
(
n
−
k
)
!

.

{\displaystyle \binom{n}{k} = \frac{n!}{k! (n-k)!}.}

 See Binomial coefficientw
Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to complex exponents.) In this generalization, the finite sum is replaced by an infinite series.

x
2

−

y
2

=
(
x
y
)
(
x
−
y
)

{\displaystyle x^2 - y^2 = (x + y)(x - y)}

x
2

y
2

=
(
x
y
i
)
(
x
−
y
i
)

{\displaystyle x^2 + y^2 = (x + yi)(x - yi)}

The polynomial remainder theoremw states that the remainder of the division of a polynomial Z(x) by the linear polynomial x-a is equal to Z(a). See Ruffini's rule.

Determining the value at Z(a) is sometimes easier if we use Horner's method (synthetic division) by writing the polynomial in the form

Z
(
x
# )

a
0

x
(

a
1

x
(

a
2

⋯
x
(

a

n
−
1

x
(

a
n

)
)
)
)
.

{\displaystyle Z(x) = a\_0 + x(a\_1 + x(a\_2 + \cdots + x(a\_{n-1} + x(a\_n)))). }

A monic polynomial is a one variable polynomial in which the leading coefficient is equal to 1.

a
0

a
1

x

a
2

x
2

⋯

a

n
−
1

x

n
−
1

1

x
n

{\displaystyle a\_0 + a\_1x + a\_2x^2 + \cdots + a\_{n-1}x^{n-1} + 1x^n}

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### Rational functions

A rational function is a function of the form

f
(
x
# )
k

(
x
−

z
1

)
(
x
−

z
2

)
⋯
(
x
−

z
n

)

(
x
−

p
1

)
(
x
−

p
2

)
⋯
(
x
−

p
m

)

=

Z
(
x
)

P
(
x
)

{\displaystyle f(x) = k{(x - z\_1)(x - z\_2)\dotsb(x - z\_n) \over (x - p\_1)(x - p\_2)\dotsb(x - p\_m)} = {Z(x) \over P(x)}}

It has n zerosw and m polesw. A pole is a value of x for which |f(x)| = infinity.

The vertical asymptotesw are the poles of the rational function.
If n<m then f(x) has a horizontal asymptote at the x axis
If n=m then f(x) has a horizontal asymptote at k.
If n>m then f(x) has no horizontal asymptote.
See also Wikipedia:Asymptote#Oblique\_asymptotes
Given two polynomials

Z
(
x
)

{\displaystyle Z(x)}

 and

P
(
x
# )
(
x
−

p
1

)
(
x
−

p
2

)
⋯
(
x
−

p
m

)

{\displaystyle P(x) = (x-p\_1)(x-p\_2) \cdots (x-p\_m)}

, where the *p**i* are distinct constants and deg *Z* < *m*, partial fractionsw are generally obtained by supposing that

Z
(
x
)

P
(
x
)

=

c
1

x
−

p
1

c
2

x
−

p
2

⋯

c
m

x
−

p
m

{\displaystyle \frac{Z(x)}{P(x)} = \frac{c\_1}{x-p\_1} + \frac{c\_2}{x-p\_2} + \cdots + \frac{c\_m}{x-p\_m}}

and solving for the *c**i* constants, by substitution, by equating the coefficients of terms involving the powers of *x*, or otherwise.
(This is a variant of the method of undetermined coefficients.)
If the degree of Z is not less than m then use long division to divide P into Z. The remainder then replaces Z in the equation above and one proceeds as before.
If

P
(
x
# )
(
x
−
p

)
m

{\displaystyle P(x) = (x-p)^m}

 then

Z
(
x
)

P
(
x
)

=

c
1

(
x
−
p
)

c
2

(
x
−
p

)
2

⋯

c
m

(
x
−
p

)
m

{\displaystyle \frac{Z(x)}{P(x)} = \frac{c\_1}{(x-p)} + \frac{c\_2}{(x-p)^2} + \cdots + \frac{c\_m}{(x-p)^m}}

A Generalized hypergeometric series is given by

∑

# x
0

c
x

{\displaystyle \sum\_{x=0} c\_x}

 where c0=1 and

c

x
1

c
x

=

Z
(
x
)

P
(
x
)

=
f
(
x
)

{\displaystyle {c\_{x+1} \over c\_x} = {Z(x) \over P(x)} = f(x)}

The function f(x) has n zeros and m poles.

Basic hypergeometric series, or hypergeometric q-series, are q-analogue generalizations of generalized hypergeometric series.
Roughly speaking a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1
We define the *q*-analog of *n*, also known as the ***q*-bracket** or ***q*-number** of *n*, to be

[
n

]
q

=

1
−

q
n

1
−
q

=

q
0

q
1

q
2

…

q

n
−
1

{\displaystyle [n]\_q=\frac{1-q^n}{1-q} = q^0 + q^1 + q^2 + \ldots + q^{n - 1}}

one may define the *q*-analog of the factorialw, known as the *q*-factorial, by

[
n

]
q

# !
[
1

]
q

⋅
[
2

]
q

⋯
[
n
−
1

]
q

⋅
[
n

]
q

{\displaystyle
[n]\_q! = [1]\_q \cdot [2]\_q \cdots [n-1]\_q \cdot [n]\_q
}

Elliptic hypergeometric series are generalizations of basic hypergeometric series.
An elliptic function is a meromorphic function that is periodic in two directions.
A generalized hypergeometric function is given by

F
(
x
# )

n

F
m

(

z
1

,
.
.
.

z
n

;

p
1

,
.
.
.

p
m

;
x

# y
0

c
y

x
y

{\displaystyle F(x) = {}\_nF\_m(z\_1,...z\_n;p\_1,...p\_m;x) = \sum\_{y=0} c\_y x^y}

So for ex (see below) we have:

c
y

=

1

y
!

,

c

y
1

c
y

=

1

y
1

.

{\displaystyle c\_y = \frac{1}{y!}, \qquad \frac{c\_{y+1}}{c\_y} = \frac{1}{y+1}.}

Back to top

## Integration and differentiation

See also: Hyperreal numberw and Implicit differentiationw
The **integralw** is a generalization of multiplication.

For example: a unit mass dropped from point x2 to point x1 will release energy.
The usual equation is is a simple multiplication:

g
r
a
v
i
t
y
⋅
(

x
2

−

x
1

# )
e
n
e
r
g
y

{\displaystyle gravity \cdot (x\_2 - x\_1) = energy}

But that equation cant be used if the strength of gravity is itself a function of x.
The strength of gravity at x1 would be different than it is at x2.
And in reality gravity really does depend on x (x is the distance from the center of the earth):

g
r
a
v
i
t
y
(
x
# )
1

/

x
2

{\displaystyle gravity(x) = 1/x^2}

 (See inverse-square laww.)
However, the corresponding Definite integralw is easily solved:

∫

x
1

x
2

g
r
a
v
i
t
y
(
x
)
⋅
d
x

{\displaystyle \int\_{x\_1}^{x\_2} gravity(x) \cdot dx}

∫

x
1

x
2

f
(
x
)
⋅
d
x

=

F
(

x
2

)
−
F
(

x
1

)

{\displaystyle \int\_{x\_1}^{x\_2} f(x) \cdot dx \quad = \quad F(x\_2)-F(x\_1)}

F(x) is called the indefinite integralw. (antiderivativew)

F
(
x
# )
∫
f
(
x
)
⋅
d
x

{\displaystyle F(x) = \int f(x) \cdot dx}

*k* and *y* are arbitrary constants:

∫
k
⋅

x
y

⋅
d
x

=

k
⋅
∫

x
y

⋅
d
x

=

k
⋅

x

y
1

y
1

{\displaystyle \int k \cdot x^y \cdot dx \quad = \quad k \cdot \int x^y \cdot dx \quad = \quad k \cdot \frac{x^{y+1}}{y+1}}

(Units (feet, mm...) behave exactly like constants.)
And most conveniently :

∫

(

f
(
x
)
g
(
x
)

)

⋅
d
# x
∫
f
(
x
)
⋅
d
x
∫
g
(
x
)
⋅
d
x

{\displaystyle \int \bigg (f(x) + g(x) \bigg) \cdot dx = \int f(x) \cdot dx + \int g(x) \cdot dx }

The integral of a function is equal to the area under the curve.
When the "curve" is a constant (in other words, k•x0) then the integral reduces to ordinary multiplication.
The **derivativew** is a generalization of division.

The derivative of the integral of f(x) is just f(x).

The derivative of a function at any point is equal to the slope of the function at that point.

f
′

(
x
# )

f
(
x
d
x
)
−
f
(
x
)

d
x

.

{\displaystyle f'(x)=\frac{f(x+dx)-f(x)}{dx}.}

The equation of the line tangent to a function at point a is

y
(
x
# )
f
(
a
)

f
′

(
a
)
(
x
−
a
)

{\displaystyle y(x) = f(a) + f'(a)(x-a)}

The Lipschitz constantw of a function is a real number for which the absolute value of the slope of the function at every point is not greater than this real number.

The derivative of f(x) where f(x) = k•xy is

f
′

(
x
# )

d
f

d
x

=

d
(
k
⋅

x
y

)

d
x

=

k
⋅

d
(

x
y

)

d
x

=

k
⋅
y
⋅

x

y
−
1

{\displaystyle f'(x) = {df \over dx} = {d(k \cdot x^y) \over dx} \quad = \quad k \cdot {d(x^y) \over dx} \quad = \quad k \cdot y \cdot x^{y-1}}

The derivative of a

k
⋅

x
0

{\displaystyle k \cdot x^0}

 is

k
⋅
0
⋅

x

−
1

{\displaystyle k \cdot 0 \cdot x^{-1}}

The integral of

x

−
1

{\displaystyle x^{-1}}

 is ln(x). See natural logw
Chain rulew for the derivative of a function of a function:

f
(
g
(
x
)

)
′

=

d
f

d
x

=

d
f

d
g

⋅

d
g

d
x

{\displaystyle f(g(x))' = \frac{df}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}}

The Chain rule for a function of 2 functions:

f
(
g
(
x
)
,
h
(
x
)

)
′

=

d
⁡
f

d
⁡
x

=

∂
f

∂
g

d
⁡
g

d
⁡
x

∂
f

∂
h

d
⁡
h

d
⁡
x

{\displaystyle f(g(x), h(x))' = \frac{\operatorname df}{\operatorname dx} = { \partial f \over \partial g}{\operatorname dg \over \operatorname dx} + {\partial f \over \partial h}{\operatorname dh \over \operatorname dx }}

 (See "partial derivatives" below)
The Product rulew can be considered a special case of the chain rulew for several variables

d
f

d
x

=

d
(
g
(
x
)
⋅
h
(
x
)
)

d
x

=

∂
(
g
⋅
h
)

∂
g

d
g

d
x

∂
(
g
⋅
h
)

∂
h

d
h

d
x

=

d
g

d
x

h
g

d
h

d
x

{\displaystyle \frac{df}{dx} = {d (g(x) \cdot h(x)) \over dx} = \frac{\partial(g \cdot h)}{\partial g}\frac{dg}{dx}+\frac{\partial (g \cdot h)}{\partial h}\frac{dh}{dx} = \frac{dg}{dx} h + g \frac{dh}{dx}}

Product rulew:

(
g
⋅
h

)
′

=

(
g
d
g
)
⋅
(
h
d
h
)
−
g
⋅
h

d
x

=

g
′

⋅
h
g
⋅

h
′

{\displaystyle (g \cdot h)' = \frac{(g+dg) \cdot (h+dh) - g \cdot h}{dx} = g' \cdot h + g \cdot h'}

 (because

d
h
⋅
d
g

{\displaystyle dh \cdot dg}

 is negligible)

(
g
⋅
h
⋅
j

)
′

=

g
′

⋅
h
⋅
j
g
⋅

h
′

⋅
j
g
⋅
h
⋅

j
′

{\displaystyle (g \cdot h \cdot j)' = g' \cdot h \cdot j + g \cdot h' \cdot j + g \cdot h \cdot j'}

General Leibniz rule:

(
g
h

)

(
n
)

=

∑

# k
0

n

(

n
k

)

g

(
n
−
k
)

h

(
k
)

{\displaystyle (gh)^{(n)}=\sum\_{k=0}^n {n \choose k} g^{(n-k)} h^{(k)}}

By the chain rule:

(

1
h

)

′

=

−
1

h
2

⋅

h
′

{\displaystyle \bigg(\frac{1}{h}\bigg)' = \frac{-1}{h^2} \cdot h'}

Therefore the Quotient rulew:

(

g
(
x
)

h
(
x
)

)

′

=

(

g
⋅

1
h

)

′

=

g
′

⋅

1
h

g
⋅

−

h
′

h
2

=

g
′

⋅
h
−
g
⋅

h
′

h
2

{\displaystyle \bigg( \frac{g(x)}{h(x)} \bigg)' = \bigg( g \cdot \frac{1}{h} \bigg)' = g' \cdot \frac{1}{h} + g \cdot \frac{-h'}{h^2} = \frac{g' \cdot h - g \cdot h'}{h^2}}

There is a chain rule for integration but the inner function must have the form

# g
a
x
c

{\displaystyle g=ax+c}

 so that its derivative

d
g

d
x

=
a

{\displaystyle \frac{dg}{dx} = a}

 and therefore

d
# x

d
g

a

{\displaystyle dx=\frac{dg}{a}}

∫
f
(
g
(
x
)
)
⋅
d
# x
∫
f
(
g
)
⋅

d
g

a

=

1
a

∫
f
(
g
)
⋅
d
g

{\displaystyle \int f(g(x)) \cdot dx = \int f(g) \cdot \frac{dg}{a} = \frac{1}{a} \int f(g) \cdot dg}

Actually the inner function can have the form

# g
a

x
y

c

{\displaystyle g=ax^y+c}

 so that its derivative

d
g

d
x

=
a
⋅
y
⋅

x

y
−
1

{\displaystyle \frac{dg}{dx} = a \cdot y \cdot x^{y-1}}

 and therefore

d
# x

d
g

a
⋅
y
⋅

x

y
−
1

{\displaystyle dx=\frac{dg}{a \cdot y \cdot x^{y-1}}}

 provided that all factors involving x cancel out.

∫

x

y
−
1

⋅
f
(
g
(
x
)
)
⋅
d
# x
∫

x

y
−
1

⋅
f
(
g
)
⋅

d
g

a
⋅
y
⋅

x

y
−
1

=

1

a
⋅
y

∫
f
(
g
)
⋅
d
g

{\displaystyle \int x^{y-1} \cdot f(g(x)) \cdot dx = \int {\color{red} x^{y-1}} \cdot f(g) \cdot \frac{dg}{a \cdot y \cdot {\color{red} x^{y-1}}} = \frac{1}{a \cdot y} \int f(g) \cdot dg}

The product rule for integration is called Integration by partsw

g
⋅

h
′

=
(
g
⋅
h

)
′

−

g
′

⋅
h

{\displaystyle g \cdot h' = (g \cdot h)' - g' \cdot h}

∫
g
⋅

h
′

⋅
d
# x
g
⋅
h
−
∫

g
′

⋅
h
⋅
d
x

{\displaystyle \int g \cdot h' \cdot dx = g \cdot h - \int g' \cdot h \cdot dx}

One can use partial fractionsw or even the Taylor seriesw to convert difficult integrals into a more manageable form.

f
(
x
)

(
x
−
1

)
2

=

a
0

(
x
−
1

)
0

a
1

(
x
−
1

)
1

⋯

a
n

(
x
−
1

)
n

(
x
−
1

)
2

{\displaystyle \frac{f(x)}{(x-1)^2} = \frac{a\_0(x-1)^0 + a\_1(x-1)^1 + \dots + a\_n(x-1)^n}{(x-1)^2}}

The fundamental theorem of Calculus is:

F
(
x
)
−
F
(
a
# )

∫
a
x

f
(
t
)

d
t

and

F
′

(
x
# )
f
(
x
)

{\displaystyle F(x) - F(a) = \int\_a^x\!f(t)\, dt \quad \text{and} \quad F'(x) = f(x)}

The fundamental theorem of calculus is just the particular case of the Leibniz integral rule:

d

d
x

(

∫

a
(
x
)

b
(
x
)

f
(
x
,
t
)

d
t

)

=
f

(

x
,
b
(
x
)

)

⋅

d

d
x

b
(
x
)
−
f

(

x
,
a
(
x
)

)

⋅

d

d
x

a
(
x
)

∫

a
(
x
)

b
(
x
)

∂

∂
x

f
(
x
,
t
)

d
t
.

{\displaystyle \frac{d}{dx} \left (\int\_{a(x)}^{b(x)}f(x,t)\,dt \right) = f\big(x,b(x)\big)\cdot \frac{d}{dx} b(x) - f\big(x,a(x)\big)\cdot \frac{d}{dx} a(x) + \int\_{a(x)}^{b(x)}\frac{\partial}{\partial x} f(x,t) \,dt.}

In calculus, a function f defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely non-increasing, or entirely non-decreasing.

A differential formw is a generalisation of the notion of a differentialw that is independent of the choice of coordinate system. *f(x,y) dx ∧ dy* is a 2-form in 2 dimensions (an area element). The derivativew operation on an *n*-form is an *n+1*-form; this operation is known as the exterior derivativew. By the generalized Stokes' theoremw, the integral of a function over the boundary of a manifoldw is equal to the integral of its exterior derivative on the manifold itself.

Back to top

### Taylor & Maclaurin series

See also: Wikipedia:Volterra series
If we know the value of a smooth functionw at x=0 (smooth means all its derivatives are continuousw) and we also know the value of all of its derivatives at x=0 then we can determine the value at any other point *x* by using the Maclaurin seriesw. ("!" means factorialw)

f
(
x
# )

a
0

x
0

a
1

x
1

a
2

x
2

a
3

x
3

⋯

where

a
n

=

f

(
n
)

(
0
)

n
!

{\displaystyle f(x) = a\_0 x^0 + a\_1 x^1 + a\_2 x^2 + a\_3 x^3 \cdots \quad \text{where} \quad a\_n = {f^{(n)}(0) \over n!}}

The proof of this is actually quite simple. Plugging in a value of *x=0* causes all terms but the first to become zero. So, assuming that such a function exists, *a0* must be the value of the function at *x=0*. Simply differentiate both sides of the equation and repeat for the next term. And so on.

Because the functions

x
n

{\displaystyle x^n}

 can be multiplied by scalars and added they therefore form an infinite dimensional vector space. (An infinite dimensional space is not a Compact space.) The function f(x) occupies a single point in that infinite dimensional space corresponding to a vector whose components are

a
n

{\displaystyle a\_n}

v

=

[

a
0

a
1

a
2

…

a

∞

]

{\displaystyle \mathbf{v} = \begin{bmatrix} a\_0 & a\_1 & a\_2 & \ldots & a\_{\infty} \end{bmatrix}}

The Taylor seriesw generalizes the Maclaurin series.

f
(
z

# k
0

∞

α
k

(
z
−

z
0

)
k

{\displaystyle f(z)=\sum\_{k=0}^\infty \alpha\_k (z-z\_0)^k}

An analytic functionw is a function whose Taylor series converges for every *z0* in its domainw; analytic functions are infinitely differentiablew.
Any vector *g* = (*z*0, α0, α1, ...) is a germ if it represents a power series of an analytic functionw around *z*0 with some radius of convergence *r* > 0.
The set of germs

G

{\displaystyle \mathcal G}

 is a Riemann surfacew.
Riemann surfaces are the objects on which multi-valued functions become single-valued.
A connected component of

G

{\displaystyle \mathcal G}

 (i.e., an equivalence class) is called a *sheaf*.
We can easily determine the Maclaurin series expansion of the exponential functionw

e
x

{\displaystyle e^x}

 because it is equal to its own derivative.

e
x

=

∑

# n
0

∞

x
n

n
!

=

x
0

0
!

x
1

1
!

x
2

2
!

x
3

3
!

x
4

4
!

⋯

{\displaystyle e^x = \sum\_{n = 0}^{\infty} {x^n \over n!} = {x^0 \over 0!} + {x^1 \over 1!} + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots}

The above holds true even if x is a matrix. See Matrix exponential
And likewise for cos(x)w and sin(x)w because cosine is the derivative of sine which is the derivative of -cosine

cos
⁡
# x

x
0

0
!

−

x
2

2
!

x
4

4
!

−

x
6

6
!

⋯

{\displaystyle

\cos x = \frac{x^0}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots

}

sin
⁡
# x

x
1

1
!

−

x
3

3
!

x
5

5
!

−

x
7

7
!

⋯

{\displaystyle

\sin x = \frac{x^1}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots

}

It then follows that

e

i
x

=
cos
⁡
x
i
sin
⁡
# x
cis
⁡
x

{\displaystyle e^{ix}=\cos x+i\sin x=\operatorname{cis} x}

 and therefore

e

i
π

=
−
1
i
⋅
0

{\displaystyle e^{i \pi}=-1 + i\cdot0}

 See Euler's formulaw

*x* is the angle in radians.
This makes the equation for a circle in the complex plane, and by extension sine and cosine, extremely simple and easy to work with especially with regard to differentiation and integration.

d
(

e

i
⋅
k
⋅
t

)

d
t

=
i
⋅
k
⋅

e

i
⋅
k
⋅
t

{\displaystyle \frac{d(e^{i \cdot k \cdot t})}{dt} = i \cdot k \cdot e^{i \cdot k \cdot t}}

For sine waves differentiation and integration are replaced with multiplication and division. Calculus is replaced with algebra. Therefore any expression that can be represented as a sum of sine waves can be easily differentiated or integrated.
Back to top

### Fourier Series

The Maclaurin series cant be used for a discontinuous function like a square wave because it is not differentiable. (Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. See Generalized function.)

But remarkably we can use the Fourier seriesw to expand it or any other periodic functionw into an infinite sum of sine waves each of which is fully differentiablew!

f
(
t
# )

a
0

2

∑

# n
1

∞

[

a
n

cos
⁡

(

n
t

)

b
n

sin
⁡

(

n
t

)

]

{\displaystyle
f(t) = \frac{a\_0}{2} + \sum\_{n=1}^\infty \left[a\_n\cos\left(nt\right)+b\_n\sin\left(nt\right)\right]
}

a
n

=

2
p

∫

t
0

t
p

f
(
t
)
⋅
cos
⁡

(

2
π
n
t

p

)

d
t

{\displaystyle a\_n = \frac{2}{p}\int\_{t\_0}^{t\_p} f(t)\cdot \cos\left(\tfrac{2\pi nt}{p}\right)\ dt}

b
n

=

2
p

∫

t
0

t
p

f
(
t
)
⋅
sin
⁡

(

2
π
n
t

p

)

d
t

{\displaystyle b\_n = \frac{2}{p}\int\_{t\_0}^{t\_p} f(t)\cdot \sin\left(\tfrac{2\pi nt}{p}\right)\ dt}

A square wave consists of all odd frequencies with the amplitude of each frequency being

1
f

{\displaystyle \frac{1}{f}}

.

A dirac comb consists of all integer frequencies with the amplitude of each frequency being 1.

The reason this works is because sine and cosine are orthogonal functions.
Two vectors are said to be orthogonal when:

u

⋅

v

=
0

{\displaystyle \mathbf{u} \cdot \mathbf{v} = 0}

or more generally when:

β
(

u
,
v

# )
0

{\displaystyle \beta(\mathbf{u,v}) = 0}

Two functions are said to be orthogonal when:

⟨
f
,
g
# ⟩
∫

f
(
x
)

¯

g
(
x
)

d
# x

{\displaystyle \langle f,g \rangle = \int \overline{f(x)}g(x)\,dx = 0.}

That means that multiplying any 2 sine waves of frequency n and frequency m and integrating over one period will always equal zero unless n=m.
See the graph of sin2(x) to the right.

sin
⁡
m
x
⋅
sin
⁡
n
# x

cos
⁡
(
m
−
n
)
x
−
cos
⁡
(
m
n
)
x

2

{\displaystyle \sin mx \cdot \sin nx = \frac{\cos (m - n)x - \cos (m+n) x}{2}}

See Amplitude\_modulation
And of course ∫ fn\*(f1+f2+f3+...) = ∫ (fn\*f1) + ∫ (fn\*f2) + ∫ (fn\*f3) +...
The complex form of the Fourier series uses complex exponentials instead of sine and cosine and uses both positive and negative frequencies (clockwise and counter clockwise) whose imaginary parts cancel.
The complex coefficients encode both amplitude and phase and are complex conjugates of each other.

F
(
ν
# )

F

{
f
# }

∫

R

n

f
(
x
)

e

−
2
π
i
x
⋅
ν

d

x

{\displaystyle F(\nu) = \mathcal{F}\{f\} = \int\_{\mathbb{R}^n} f(x) e^{-2 \pi i x\cdot\nu} \, \mathrm{d}x }

where the *dot* between *x* and *ν* indicates the inner productw of **R***n*.
A 2 dimensional Fourier series is used in video compression.
A discrete Fourier transform can be computed very efficiently by a fast Fourier transform (FFT).
The FFT has been described as "the most important numerical algorithm of our lifetime"
In mathematical analysis, many generalizations of Fourier series have proven to be useful.
They are all special cases of decompositions over an orthonormal basis of an inner product space.
Spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via Fourier series.
Spherical harmonics are basis functions for SO(3). See Laplace seriesw.
Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.
Every quadratic polynomial can be written as *a*1+*bt*+*ct*2, that is, as a linear combination of the basis functions 1, *t*, and *t*2.
Back to top

### Transforms

Fourier transformsw generalize Fourier series to nonperiodic functions like a single pulse of a square wave.

The more localized in the time domain (the shorter the pulse) the more the Fourier transform is spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principlew.

Using the Fourier transform we can determine that the Dirac delta functionw consists of all frequencies with the amplitude of each frequency being 1.

G
(
ω
# )

F

{
f
(
t
)
# }

∫

−
∞

∞

f
(
t
)

e

−
i
ω
t

d
t

{\displaystyle G(\omega)=\mathcal{F}\{f(t)\}=\int\_{-\infty}^\infty f(t) e^{-i\omega t}dt}

Laplace transformsw generalize Fourier transforms to complex frequency

# s
σ
i
ω

{\displaystyle s=\sigma+i\omega}

.
Complex frequency includes a term corresponding to the amount of damping.
See http://www.dspguide.com/CH32.PDF.

F
(
s
# )

L

{
f
(
t
)
# }

∫
0
∞

f
(
t
)

e

−
σ
t

e

−
i
ω
t

d
t

{\displaystyle F(s)=\mathcal{L}\{f(t)\}=\int\_0^\infty f(t) e^{-\sigma t}e^{-i \omega t}dt}

L

{
δ
(
t
−
a
)
# }

e

−
a
s

{\displaystyle \mathcal{L}\{ \delta(t-a) \} = e^{-as}}

, (assuming *a* > 0)

L

{

e

a
t

# }

1

s
−
a

{\displaystyle \mathcal{L}\{e^{at} \}= \frac{1}{s - a}}

The inverse Laplace transformw is given by

f
(
t
# )

L

−
1

{
F
# }

1

2
π
i

lim

T
→
∞

∫

γ
−
i
T

γ
i
T

F
(
s
)

e

s
t

d
s
,

{\displaystyle f(t) = \mathcal{L}^{-1} \{F\} = \frac{1}{2\pi i}\lim\_{T\to\infty}\int\_{\gamma-iT}^{\gamma+iT}F(s)e^{st}\,ds,}

where the integration is done along the vertical line Re(*s*) = *γ* in the complex planew such that *γ* is greater than the real part of all singularities of *F*(*s*) and *F*(*s*) is bounded on the line, for example if contour path is in the region of convergence.
If all singularities are in the left half-plane, or *F*(*s*) is an entire function , then *γ* can be set to zero and the above inverse integral formula becomes identical to the inverse Fourier transform.
The Z-transform can be considered as a discrete-time equivalent of the Laplace transform. This similarity is explored in the theory of time-scale calculus.
Integral transformsw generalize Fourier transforms to other kernals (besides sinew and cosinew)
Cauchy kernel =

1

ζ
−
x

or

1

2
π
i

⋅

1

ζ
−
x

{\displaystyle \frac{1}{\zeta-x} \quad \text{or} \quad \frac{1}{2\pi i} \cdot \frac{1}{\zeta-x}}

Hilbert kernel =

c
o
t

θ
−
t

2

{\displaystyle cot\frac{\theta-t}{2}}

Poisson Kernel:
For the ball of radius r,

B

r

{\displaystyle B\_{r}}

, in **R**n, the Poisson kernel takes the form:

P
(
x
,
ζ
# )

r
2

−

x

2

r

⋅

1

ζ
−
x

n

⋅

1

ω

n

where

x
∈

B

r

{\displaystyle x\in B\_{r}}

,

ζ
∈
S

{\displaystyle \zeta\in S}

 (the surface of

B

r

{\displaystyle B\_{r}}

), and

ω
n

{\displaystyle \omega\_n}

 is the surface area of the unit n-sphere.
unit disk (r=1) in the complex plane:

K
(
x
,
ϕ
# )

1
2

−

x

2

1

⋅

1

e

i
ϕ

−
x

2

⋅

1

2
π

Dirichlet kernel

D
n

(
x

# k
−
n

n

e

i
k
x

=
1
2

∑

# k
1

n

cos
⁡
(
k
x
# )

sin
⁡

(

(

n

1
2

)

x

)

sin
⁡
(

x
2

)

≈
2
π
δ
(
x
)

{\displaystyle D\_n(x)=\sum\_{k=-n}^n

e^{ikx}=1+2\sum\_{k=1}^n\cos(kx)=\frac{\sin\left(\left(n + \frac{1}{2}\right) x \right)}{\sin(\frac{x}{2})} \approx 2\pi\delta(x)}

The convolution theorem states that

F

{
f
∗
g
# }

F

{
f
}
⋅

F

{
g
}

{\displaystyle \mathcal{F}\{f\*g\} = \mathcal{F}\{f\} \cdot \mathcal{F}\{g\}}

where

⋅

{\displaystyle \cdot}

 denotes point-wise multiplication. It also works the other way around:

F

{
f
⋅
g
# }

F

{
f
}
∗

F

{
g
}

{\displaystyle \mathcal{F}\{f \cdot g\}= \mathcal{F}\{f\}\*\mathcal{F}\{g\}}

By applying the inverse Fourier transform

F

−
1

{\displaystyle \mathcal{F}^{-1}}

, we can write:

f
∗
# g

F

−
1

{

F

{
f
}
⋅

F

{
g
}

}

{\displaystyle f\*g= \mathcal{F}^{-1}\big\{\mathcal{F}\{f\}\cdot\mathcal{F}\{g\}\big\}}

and:

f
⋅
# g

F

−
1

{

F

{
f
}
∗

F

{
g
}

}

{\displaystyle f \cdot g= \mathcal{F}^{-1}\big\{\mathcal{F}\{f\}\*\mathcal{F}\{g\}\big\}}

This theorem also holds for the Laplace transformw.

The Hilbert transform is a multiplier operator. The multiplier of *H* is σ*H*(ω) = −*i* sgn(ω) where sgn is the signum function. Therefore:

F

(
H
(
u
)
)
(
ω
# )
(
−
i

sgn
⁡
(
ω
)
)
⋅

F

(
u
)
(
ω
)

{\displaystyle \mathcal{F}(H(u))(\omega) = (-i\,\operatorname{sgn}(\omega)) \cdot \mathcal{F}(u)(\omega)}

where

F

{\displaystyle \mathcal{F}}

 denotes the Fourier transformw.

Since sgn(*x*) = sgn(2π*x*), it follows that this result applies to the three common definitions of

F

{\displaystyle \mathcal{F}}

.

By Euler's formulaw,

σ
H

(
ω

# i

e

i
π

2

,

for
ω
<
0

0
,

for
# ω
0

−
# i

e

−

i
π

2

,

for
ω
>
0

{\displaystyle \sigma\_H(\omega) = \begin{cases}

 i = e^{+\frac{i\pi}{2}}, & \text{for } \omega < 0\\

 0, & \text{for } \omega = 0\\

 -i = e^{-\frac{i\pi}{2}}, & \text{for } \omega > 0

\end{cases}}

Therefore, *H*(*u*)(*t*) has the effect of shifting the phase of the negative frequency components of *u*(*t*) by +90° (π/2 radians) and the phase of the positive frequency components by −90°.

And *i*·*H*(*u*)(*t*) has the effect of restoring the positive frequency components while shifting the negative frequency ones an additional +90°, resulting in their negation.

In electrical engineering, the convolution of one function (the input signal) with a second function (the impulse response) gives the output of a linear time-invariant system (LTI).

At any given moment, the output is an accumulated effect of all the prior values of the input function

Back to top

### Differential equations

See also: Variation of parameters

Simple harmonic motion of a mass on a spring is a second-order linear ordinary differential equationw.

F
o
r
c
# e
m
a
s
s
∗
a
c
# c
m

d

2

x

d

t
2

=
−
k
x
,

{\displaystyle Force = mass\*acc = m\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -kx,}

where *m* is the inertial mass, *x* is its displacement from the equilibrium, and *k* is the spring constant.

Solving for x produces

x
(
t
# )
A
cos
⁡

(

ω
t
−
φ

)

,

{\displaystyle x(t) = A\cos\left(\omega t - \varphi\right),}

*A* is the amplitude (maximum displacement from the equilibrium position),

# ω
2
π
# f

k

/

m

{\displaystyle \omega = 2\pi f = \sqrt{k/m}}

 is the angular frequencyw, and *φ* is the phase.

Energy passes back and forth between the potential energy in the spring and the kinetic energy of the mass.

The important thing to note here is that the frequency of the oscillation depends only on the mass and the stiffness of the spring and is totally independent of the amplitude.

That is the defining characteristic of resonance.

Kirchhoff's voltage law states that the sum of the emfs in any closed loop of any electronic circuit is equal to the sum of the voltage drops in that loop.

V
(
t
# )

V
R

V
L

V
C

{\displaystyle V(t) = V\_R + V\_L + V\_C}

V is the voltage, R is the resistance, L is the inductance, C is the capacitance.

V
(
t
# )
R
I
(
t
)
L

d
I
(
t
)

d
t

1
C

∫

0

t

I
(
τ
)

d
τ

{\displaystyle V(t) = RI(t) + L \frac{dI(t)}{dt} + \frac{1}{C} \int\_{0}^t I(\tau)\, d\tau}

I = dQ/dt is the current.

It makes no difference whether the current is a small number of charges moving very fast or a large number of charges moving slowly.

In reality the latter is the case.

If V(t)=0 then the only solution to the equation is the transient response which is a rapidly decaying sine wave with the same frequency as the resonant frequency of the circuit.

Like a mass (inductance) on a spring (capacitance) the circuit will resonate at one frequency.
Energy passes back and forth between the capacitor and the inductor with some loss as it passes through the resistor.
If V(t)=sin(t) from -∞ to +∞ then the only solution is a sine wave with the same frequency as V(t) but with a different amplitude and phase.

If V(t) is zero until t=0 and then equals sin(t) then I(t) will be zero until t=0 after which it will consist of the steady state response plus a transient response.

From Wikipedia:Characteristic equation (calculus):

Starting with a linear homogeneous differential equation with constant coefficients

a

n

,

a

n
−
1

,
…
,

a

1

,

a

0

{\displaystyle a\_{n}, a\_{n-1}, \ldots , a\_{1}, a\_{0}}

,

a

n

y

(
n
)

a

n
−
1

y

(
n
−
1
)

⋯

a

1

y
′

a

0

# y
0

{\displaystyle a\_{n}y^{(n)} + a\_{n-1}y^{(n-1)} + \cdots + a\_{1}y^\prime + a\_{0}y = 0}

it can be seen that if

y
(
x
# )

e

r
x

{\displaystyle y(x) = e^{rx} \, }

, each term would be a constant multiple of

e

r
x

{\displaystyle e^{rx} \, }

. This results from the fact that the derivative of the exponential functionw

e

r
x

{\displaystyle e^{rx} \, }

 is a multiple of itself. Therefore,

y
′

=
r

e

r
x

{\displaystyle y' = re^{rx} \, }

,

y
″

=

r

2

e

r
x

{\displaystyle y'' = r^{2}e^{rx} \, }

, and

y

(
n
)

=

r

n

e

r
x

{\displaystyle y^{(n)} = r^{n}e^{rx} \, }

 are all multiples. This suggests that certain values of

r

{\displaystyle r \, }

 will allow multiples of

e

r
x

{\displaystyle e^{rx} \, }

 to sum to zero, thus solving the homogeneous differential equation. In order to solve for

r

{\displaystyle r \, }

, one can substitute

# y

e

r
x

{\displaystyle y = e^{rx} \, }

 and its derivatives into the differential equation to get

a

n

r

n

e

r
x

a

n
−
1

r

n
−
1

e

r
x

⋯

a

1

r

e

r
x

a

0

e

r
x

=
0

{\displaystyle a\_{n}r^{n}e^{rx} + a\_{n-1}r^{n-1}e^{rx} + \cdots + a\_{1}re^{rx} + a\_{0}e^{rx} = 0}

Since

e

r
x

{\displaystyle e^{rx} \, }

 can never equate to zero, it can be divided out, giving the characteristic equation

a

n

r

n

a

n
−
1

r

n
−
1

⋯

a

1

r

a

0

=
0

{\displaystyle a\_{n}r^{n} + a\_{n-1}r^{n-1} + \cdots + a\_{1}r + a\_{0} = 0}

By solving for the roots,

r

{\displaystyle r \, }

, in this characteristic equation, one can find the general solution to the differential equation. For example, if

r

{\displaystyle r \, }

 is found to equal to 3, then the general solution will be

y
(
x
# )
c

e

3
x

{\displaystyle y(x) = ce^{3x} \, }

, where

c

{\displaystyle c \,}

 is an arbitrary constantw.

Back to top

## Partial derivatives

See also: Currying
**Partial derivatives** and **multiple integrals** generalize derivatives and integrals to multiple dimensions.

The partial derivative with respect to one variable

∂
f
(
x
,
y
)

∂
x

{\displaystyle \frac{\part f(x,y)}{\part x}}

 is found by simply treating all other variables as though they were constants.

The "partial" integral can be taken with respect to *x* (treating *y* as constant, in a similar manner to partial differentiation):

∫

∂
z

∂
x

d
# x
z
g
(
y
)

{\displaystyle \int \frac{\partial z}{\partial x} \,dx = z + g(y)}

Here, the "constant" of integration is no longer a constant, but instead a function of all the variables of the original function except *x*. The reason for this is that all the other variables are treated as constant when taking the partial derivative, so any function which does not involve

x

{\displaystyle x}

 will disappear when taking the partial derivative, and we have to account for this when we take the antiderivative.
Let f(x, y, z) be a scalar functionw (for example electric potential energy or temperature).

A 2 dimensional example of a scalar function would be an elevation map.
(Contour lines of an elevation map are an example of a level set.)

The total derivativew of *f*(*x*(*t*), *y*(*t*)) with respect to t is

d
⁡
f

d
⁡
t

=

∂
f

∂
x

d
⁡
x

d
⁡
t

∂
f

∂
y

d
⁡
y

d
⁡
t

{\displaystyle \frac{\operatorname df}{\operatorname dt} = { \partial f \over \partial x}{\operatorname dx \over \operatorname dt} + {\partial f \over \partial y}{\operatorname dy \over \operatorname dt }}

And the differentialw is

d
⁡
# f

∂
f

∂
x

d
⁡
x

∂
f

∂
y

d
⁡
y
.

{\displaystyle \operatorname df = { \partial f \over \partial x}\operatorname dx + {\partial f \over \partial y} \operatorname dy .}

Back to top

### Gradient of scalar field

The **Gradientw** of f(x, y, z) is a vector field whose value at each point is a vector (technically its a covectorw because it has units of distance−1) that points "downhill" with a magnitude equal to the slopew of the function at that point.

You can think of the gradient as how much the function changes per unit distance. That sounds like the gradient would therefore be a covector and indeed some people say it is a covector. But those people are wrong. It is a vector. Maybe it should be a covector but it **is** a vector. Its dual is a covector.

The gradient of temperature gives heat flow.

grad
⁡
(
f
# f

∂
f

∂
x

i

∂
f

∂
y

j

∂
f

∂
z

k

=

F

{\displaystyle \operatorname{grad}(f) = \nabla f = \frac{\partial f}{\partial x} \mathbf{i} +

\frac{\partial f}{\partial y} \mathbf{j} +

\frac{\partial f}{\partial z} \mathbf{k} = \mathbf{F}}

For static (unchanging) fields the Gradient of the electric potential is the electric fieldw itself. Image to the right shows the potential of a single point charge.

Its gradient gives the electric field which is shown in the 2 images below. In the image on the left the field strength is proportional to the length of the vectors. In the image on the right the field strength is proportional to the density of the flux lines. The image is 2 dimensional and therefore the flux density in the image follows an inverse first power law but in reality the field lines from a real proton or electron spread outward in 3 dimensions and therefore follow an inverse square law. Inverse square means that at twice the distance the field is four times weaker.

The field of 2 point charges is simply the linear sum of the separate charges.

#### Advanced topic

Strictly speaking what we are doing here is finding the derivative **with respect to the Cartesian coordinate system**. In flat Euclidean space the result is a vector that does not depend on which particular Cartesian coordinate system we use so we dont even bother to specify which one we used.

On a manifold things are not so simple. On a manifold we dont find the derivative with respect to a coordinate system. We find the derivative with respect to a vector (i.e. directional derivative) or a vector field (Lie derivative) on that manifold.

Back to top

### Divergence

The **Divergencew** of a vector field is a scalar.

The divergence of the electric field is non-zero wherever there is electric chargew and zero everywhere else.

Field lines begin and end at charges because the charges create the electric field.

div

F

=

∇
⋅

F

=

(

∂

∂
x

e

x

,

∂

∂
y

e

y

,

∂

∂
z

e

z

)

⋅
(

F
x

e

x

,

F
y

e

y

,

F
z

e

z

# )

∂

F
x

∂
x

e

x

⋅

e

x

∂

F
y

∂
y

e

y

⋅

e

y

∂

F
z

∂
z

e

z

⋅

e

z

{\displaystyle \operatorname{div}\,\mathbf{F} = {\color{red} \nabla\cdot\mathbf{F} }
 = \left(
\frac{\partial}{\partial x} \mathbf{e}\_x,
\frac{\partial}{\partial y} \mathbf{e}\_y,
\frac{\partial}{\partial z} \mathbf{e}\_z
\right)
\cdot (F\_x \mathbf{e}\_x,F\_y \mathbf{e}\_y,F\_z \mathbf{e}\_z)
 = \frac{\partial F\_x}{\partial x} \mathbf{e}\_x \cdot \mathbf{e}\_x
+\frac{\partial F\_y}{\partial y} \mathbf{e}\_y \cdot \mathbf{e}\_y
+\frac{\partial F\_z}{\partial z} \mathbf{e}\_z \cdot \mathbf{e}\_z
}

The **Laplacianw** is the divergence of the gradient of a function:

Δ
# f

∇
2

# f
(
∇
⋅
∇
)
# f

∂
2

f

∂

x
2

∂
2

f

∂

y
2

∂
2

f

∂

z
2

.

{\displaystyle
\Delta f = \nabla^2 f = (\nabla \cdot \nabla) f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.
}

elliptic operators generalize the Laplacian.
Back to top

### Curl

See also: Biot–Savart law
The **Curlw** of a vector field describes how much the vector field is twisted.

(The field may even go in circles.)

The curl at a certain point of a magnetic fieldw is the currentw vector at that point because current creates the magnetic fieldw.

In 3 dimensions the dual of the current vector is a bivector.

curl
(

F

# )

∇
×

F

=

e
x

e
y

e
z

∂

∂
x

∂

∂
y

∂

∂
z

F
x

F
y

F
z

{\displaystyle \text{curl} (\mathbf{F}) = {\color{blue} \nabla \times \mathbf{F} } = \begin{vmatrix} \mathbf{e\_x} & \mathbf{e\_y} & \mathbf{e\_z} \\
{\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\
 F\_x & F\_y & F\_z \end{vmatrix}
}

curl
(

F

# )

(

∂

F
z

∂
y

−

∂

F
y

∂
z

)

e
x

(

∂

F
x

∂
z

−

∂

F
z

∂
x

)

e
y

(

∂

F
y

∂
x

−

∂

F
x

∂
y

)

e
z

{\displaystyle \text{curl}( \mathbf{F}) = \left(\frac{\partial F\_z}{\partial y} - \frac{\partial F\_y}{\partial z}\right) \mathbf{e\_x} + \left(\frac{\partial F\_x}{\partial z} - \frac{\partial F\_z}{\partial x}\right) \mathbf{e\_y} + \left(\frac{\partial F\_y}{\partial x} - \frac{\partial F\_x}{\partial y}\right) \mathbf{e\_z}
}

In 2 dimensions this reduces to a single scalar

curl
(

F

# )

(

∂

F
y

∂
x

−

∂

F
x

∂
y

)

{\displaystyle \text{curl}( \mathbf{F}) = \left(\frac{\partial F\_y}{\partial x} - \frac{\partial F\_x}{\partial y}\right)}

The curl of the gradient of any scalar field is always zero.

See also: differential forms.

Back to top

#### Dual of the curl

rot
(

F

# )

∇
∧

F

=

k

j

i

∂

∂
x

∂

∂
y

∂

∂
z

F
x

F
y

F
z

=

e

y
z

e

z
x

e

x
y

∂

∂
x

∂

∂
y

∂

∂
z

F
x

F
y

F
z

=

(

∂

F
z

∂
y

−

∂

F
y

∂
z

)

e

y
z

(

∂

F
x

∂
z

−

∂

F
z

∂
x

)

e

z
x

(

∂

F
y

∂
x

−

∂

F
x

∂
y

)

e

x
y

{\displaystyle
\begin{align}
\text{rot} (\mathbf{F}) = {\color{blue} \nabla \wedge \mathbf{F} } &=
\begin{vmatrix} \mathbf{k} & \mathbf{j} & \mathbf{i} \\
{\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\
 F\_x & F\_y & F\_z
\end{vmatrix} =

\begin{vmatrix} \mathbf{e\_{yz}} & \mathbf{e\_{zx}} & \mathbf{e\_{xy}} \\
{\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\
 F\_x & F\_y & F\_z
\end{vmatrix}
\\
&=

{\color{green}
\left(
\frac{\partial F\_z}{\partial y} -
\frac{\partial F\_y}{\partial z}
\right) \mathbf{e\_{yz}}} +
{\color{green}
\left(
\frac{\partial F\_x}{\partial z} -
\frac{\partial F\_z}{\partial x}
\right) \mathbf{e\_{zx}}} +
{\color{green}
\left(
\frac{\partial F\_y}{\partial x} -
\frac{\partial F\_x}{\partial y}
\right) \mathbf{e\_{xy}}}

\end{align}
}

Back to top

### Green's theorem

The line integralw over the vector function

F
→

(
r
)

{\displaystyle \vec F(r)}

 along the path

r
(
t
)

{\displaystyle r(t)}

 is

∫
a
b

F
→

⋅

d
r

→

=

∫
a
b

F
→

(
r
(
t
)
)
⋅

r
′

(
t
)
d
t

{\displaystyle \int\limits\_a^b\vec F\cdot\vec{dr}=\int\limits\_a^b\vec F(r(t))\cdot r'(t)dt}

Green's theoremw states that if you want to know how many field lines exit a region then you can either count how many lines cross the boundary (perform a line integral) or you can simply count the number of charges (or the amount of current) within that region. See Divergence theoremw.

S

{\displaystyle {\scriptstyle S }}

F
→

⋅

d

s
→

=

∭
D

∇
⋅

F
→

d

# V

∭
D

∇
2

f

d

V

{\displaystyle \vec{F} \cdot \ \mathrm{d} \vec{s} = \iiint\_D \nabla \cdot \vec{F} \,\mathrm{d}V = \iiint\_D \nabla^2 f \,\mathrm{d}V}

In 2 dimensions this is

∮
S

F
→

⋅

n
→

d

# s

∬
D

∇
⋅

F
→

d

# A

∬
D

∇
2

f

d

A

{\displaystyle \oint\_S \vec{F} \cdot \vec{n} \ \mathrm{d} s = \iint\_D \nabla \cdot \vec{F} \ \mathrm{d} A= \iint\_D \nabla^2 f \ \mathrm{d} A}

You can think of each field line as ending in a single unit of charge.

A version of Green's theorem also works for curl.

Green's theorem is perfectly obvious when dealing with vector fields but is much less obvious when applied to complex valued functions in the complex plane.

See also Kelvin–Stokes theoremw

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### The complex plane

Highly recomend: Fundamentals of complex analysis with applications to engineering and science by Saff and Snider
External link: http://www.solitaryroad.com/c606.html
The formula for the derivative of a complex function *f* at a point *z0* is the same as for a real function:

f
′

(

z
0

# )

lim

z
→

z
0

f
(
z
)
−
f
(

z
0

)

z
−

z
0

.

{\displaystyle
f'(z\_0) = \lim\_{z \to z\_0} {f(z) - f(z\_0) \over z - z\_0 }.
}

Every complex function can be written in the form

f
(
z
# )
f
(
x
i
y
# )

f
x

(
x
,
y
)
i

f
y

(
x
,
y
)

{\displaystyle f(z)=f(x+iy)=f\_x(x,y)+i f\_y(x,y)}

Because the complex plane is two dimensional, z can approach z0 from an infinite number of different directions.

However, if within a certain region, the function f is holomorphicw (that is, complex differentiablew) then, within that region, it will only have a single derivative whose value does not depend on the direction in which z approaches z0 despite the fact that fx and fy each have 2 partial derivatives. One in the x and one in the y direction..

d
f

d
z

=

∂

f
x

∂
x

i

∂

f
y

∂
x

=

∂

f
y

∂
y

−
i

∂

f
x

∂
y

=

∂

f
x

∂
x

−
i

∂

f
x

∂
y

=

∂

f
y

∂
y

i

∂

f
y

∂
x

{\displaystyle
{df \over dz} \quad = \quad {\part f\_x \over \part x} + i {\part f\_y \over \part x} \quad = \quad {\part f\_y \over \part y} - i {\part f\_x \over \part y} \quad = \quad {\part f\_x \over \part x} - i {\part f\_x \over \part y} \quad = \quad {\part f\_y \over \part y} + i {\part f\_y \over \part x}
}

d
2

f

d

z
2

=

∂
2

f
x

∂

x
2

i

∂
2

f
y

∂

x
2

=

∂
2

f
y

∂
y
∂
x

−
i

∂
2

f
x

∂
y
∂
x

{\displaystyle {d^2f \over dz^2} \quad = \quad {\part^2 f\_x \over \part x^2} + i {\part^2 f\_y \over \part x^2} \quad = \quad {\part^2 f\_y \over \part y \part x} - i {\part^2 f\_x \over \part y \part x}}

This is only possible if the Cauchy–Riemann conditionsw are true.

∂

f
x

∂
x

=

∂

f
y

∂
y

,

∂

f
y

∂
x

=
−

∂

f
x

∂
y

{\displaystyle
\frac{\part f\_x}{\part x}=\frac{\part f\_y}{\part y}\ ,\ \quad \frac{\part f\_y}{\part x}=-\frac{\part f\_x}{\part y}
}

An entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane.

As with real valued functions, a line integral of a holomorphic function depends only on the starting point and the end point and is totally independant of the path taken.

∫
f
(
z
)
⋅
d
# z
∫
(

f
x

⋅
d
x
−

f
y

⋅
d
y
)
i
∫
(

f
y

⋅
d
x

f
x

⋅
d
y
)

{\displaystyle \int f(z) \cdot dz = \int (f\_x \cdot dx - f\_y \cdot dy) + i \int (f\_y \cdot dx + f\_x \cdot dy)}

∫
f
(
z
)
⋅
d
# z
F
(
z
# )

∫
0
t

f
(
z
(
t
)
)
⋅

d
z

d
t

⋅
d
t

{\displaystyle \int f(z) \cdot dz = F(z) = \int\_0^t f(z(t)) \cdot \frac{dz}{dt} \cdot dt}

∫
a
b

f
(
z
)
⋅
d
# z
F
(
b
)
−
F
(
a
)

{\displaystyle \int\_a^b f(z) \cdot dz = F(b) - F(a)}

The starting point and the end point for any loop are the same. This, of course, implies Cauchy's integral theoremw for any holomorphic function f:

∮
f
(
z
)

d
# z
∬

(

−
∂

f
x

∂
y

−
∂

f
y

∂
x

)

d
x

d
y
i
∬

(

∂

f
x

∂
x

−
∂

f
y

∂
y

)

d
x

d
# y
0

{\displaystyle
\oint f(z) \, dz =
\iint
\left( \frac{- \partial f\_x}{\partial y} + \frac{- \partial f\_y}{\partial x} \right) dx \, dy +
i
\iint
\left( \frac{\partial f\_x}{\partial x} + \frac{- \partial f\_y}{\partial y} \right) \, dx \, dy = 0
}

∮
f
(
z
)

d
# z
∬

(

∇
×

f
¯

i

∇
⋅

f
¯

)

d
x

d
# y
0

{\displaystyle
\oint f(z) \, dz =
\iint \left(
{\color{blue} \nabla \times \bar{f}} +
 i {\color{red} \nabla \cdot \bar{f}}
 \right) \, dx \, dy = 0
}

Therefore curl and divergence must both be zero for a function to be holomorphic.

Green's theoremw for functions (not necessarily holomorphic) in the complex plane:

∮
f
(
z
)

d
# z
2
i
∬

(

d
f

/

d

z
¯

)

d
x

d
# y
i
∬

(

∇
f

)

d
x

d
# y
i
∬

(

1

∂
f

∂
x

i

∂
f

∂
y

)

d
x

d
y

{\displaystyle
\oint f(z) \, dz =
2i \iint \left( df/d\bar{z} \right) \, dx \, dy =
i \iint \left( \nabla f \right) \, dx \, dy =
i \iint \left( 1 {\partial f \over \partial x} +
i {\partial f \over \partial y} \right) \, dx \, dy
}

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Next section: Intermediate mathematics/Discrete mathematics

# See also

* Introductory mathematics
* Advanced mathematics
* Impostor syndrome
* List of free mathematics books

# External links

* MIT open courseware
* Cheat sheets
* http://mathinsight.org
* https://math.stackexchange.com
* https://www.eng.famu.fsu.edu/~dommelen/quantum/style\_a/IV.\_Supplementary\_Informati.html
* http://www.sosmath.com
* https://webhome.phy.duke.edu/~rgb/Class/intro\_math\_review/intro\_math\_review/node1.html
* Wikiversity:Mathematics
* w:c:4chan-science:Mathematics

# References

1. ↑ Andrew Marx, *Shortcut Algebra I: A Quick and Easy Way to Increase Your Algebra I Knowledge and Test Scores*, Publisher Kaplan Publishing, 2007, ISBN 1419552880, 9781419552885, 288 pages, page 51
2. ↑ Wikipedia:Multiplicity (mathematics)
3. ↑ Wikipedia:Binomial theorem
4. ↑ Wikipedia:Partial fraction decomposition
5. ↑ Wikipedia:Basic hypergeometric series
6. ↑ Wikipedia:q-analog

ex = y = dy/dx

dx = dy/y = 1/y \* dy

∫ (1/y)dy = ∫ dx = x = ln(y)
8. ↑ Wikipedia:Product rule
9. ↑ Wikipedia:Monotonic function
10. ↑ Wikipedia:Generalized Fourier series
11. ↑ Wikipedia:Spherical harmonics
12. ↑ Wikipedia:Inverse Laplace transform
13. ↑ Wikipedia:Z-transform
14. ↑ http://mathworld.wolfram.com/PoissonKernel.html
15. ↑ Wikipedia:Convolution theorem
16. ↑ Wikipedia:RLC circuit
17. ↑ Wikipedia:Total derivative

Purge