calculation of Riemann–Stieltjes integral

calculation of Riemann–Stieltjes integral

•

If

f

is defined on

[

a

,

b

]

and

g

is a

constant function

, then

∫

a

b

f

⁢

𝑑

g

=

0

.

•

Let

f

be

continuous

on

[

a

,

b

]

,

a

<

c

<

b

and

g

the

step function

defined as

g

⁢

(

x

)

=

k

for

⁢

x

<

c

,

g

⁢

(

x

)

=

k

+

α

for

⁢

x

>

c

.

Then

∫

a

b

f

⁢

𝑑

g

=

f

⁢

(

c

)

⋅

α

.

•

Let

f

be continuous on

[

a

,

b

]

,

a

<

c

<

b

and the

function

g

be otherwise continuous but have in

x

=

c

a step of magnitude

α

.  Then

g

is sum of a

continuous function

g

*

and a step function

h

⁢

(

x

)

=

0

for

⁢

x

<

c

,

h

⁢

(

x

)

=

α

for

⁢

x

>

c

,

and one has

∫

a

b

f

⁢

𝑑

g

=

∫

a

b

f

⁢

d

⁢

(

g

*

+

h

)

=

∫

a

b

f

⁢

𝑑

g

*

+

∫

a

b

f

⁢

𝑑

h

=

∫

a

b

f

⁢

𝑑

g

*

+

f

⁢

(

c

)

⋅

α

.

•

Suppose that

g

can be expressed in the form

g

=

g

*

+

h

where

g

*

is continuous and

h

a step function having an at most

denumerable

amount of steps

α

i

in respectively the same points

c

i

on the

interval

[

a

,

b

]

as the function

g

.  If

f

is Riemann–Stieltjes

integrable

on

[

a

,

b

]

,  then

∫

a

b

f

⁢

𝑑

g

=

∫

a

b

f

⁢

𝑑

g

*

+

∑

i

f

⁢

(

c

i

)

⋅

α

i

.

(1)

•

Suppose that

g

=

g

*

+

h

(as above) has a finite amount of steps

α

i

in the points

c

i

of the interval

[

a

,

b

]

but

f

does not have same-sided

discontinuities

as

g

in any of those points.  Then

f

is Riemann–Stieltjes integrable on the interval and the equation (1) is true.

Example.

Find the value of the Riemann–

Stieltjes integral

I

:=

∫

-

3

6

(

x

-

⌊

x

⌋

)

⁢

𝑑

g

⁢

(

x

)

where the integrand

f

is the

mantissa function

and the

integrator

g

defined by

g

⁢

(

x

)

:=

{

-

x

2

for

⁢

x

≦

-

2

,

x

for

-

2

<

x

≦

3

,

2

⁢

x

+

1

⁢

for

⁢

x

>

3

.

Now,

f

is from the left

discontinuous

at every integer, but

g

is

bounded

and only discontinuous from the right at

-

2

and 3.  By the above last item,

f

is Riemann–Stieltjes integrable with respect to

g

on

[

-

3

,

6

]

.  We can set

g

=

g

*

+

h

where

g

*

is continuous and the step function

h

has the step of 2 at

-

2

and the step of 4 at 3.  Using (1) we get

I

=

∫

-

3

6

f

⁢

𝑑

g

*

+

f

⁢

(

-

2

)

⋅

2

+

f

⁢

(

3

)

⋅

4

=

∑

i

=

-

3

5

∫

i

i

+

1

f

⁢

(

x

)

⁢

g

′

⁢

(

x

)

⁢

𝑑

x

+

0

⋅

2

+

0

⋅

4

=

∫

-

3

-

2

(

x

+

3

)

⁢

(

-

2

⁢

x

)

⁢

𝑑

x

+

∫

-

2

-

1

(

x

+

2

)

⋅

1

⁢

𝑑

x

+

∫

-

1

0

(

x

+

1

)

⋅

1

⁢

𝑑

x

+

∫

0

1

x

⋅

1

⁢

𝑑

x

+

∫

1

2

(

x

-

1

)

⋅

1

⁢

𝑑

x

+

∫

2

3

(

x

-

2

)

⋅

1

⁢

𝑑

x

+

∫

3

4

(

x

-

3

)

⋅

2

⁢

𝑑

x

+

∫

4

5

(

x

-

4

)

⋅

2

⁢

𝑑

x

+

∫

5

6

(

x

-

5

)

⋅

2

⁢

𝑑

x

=

47

6

.

Title

calculation of Riemann–Stieltjes integral

Canonical name

CalculationOfRiemannStieltjesIntegral

Date of creation

2013-03-22 18:55:09

Last modified on

2013-03-22 18:55:09

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Topic

Classification

msc 26A42

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