1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Anastaziya [24]
3 years ago
7

a) Γ(n) =∫[infinity]0tn−1e−tdta)Show that Γ(n+ 1) =nΓ(n) using integration by parts.b) Show that Γ(n+ 1) =n! , wherenis a positi

ve integer, by repeatedly applying the result in parta)c) Compute Γ(5).d) Prove what the value of Γ(12) should be by evaluating the gamma function forn= 1/2 and usingthe fact that∫[infinity]0e−u2du=√π2.
Mathematics
1 answer:
Nataliya [291]3 years ago
7 0

\Gamma(n)=\displaystyle\int_0^\infty t^{n-1}e^{-t}\,\mathrm dt

a. Integrate by parts by taking

u=t^{n-1}\implies\mathrm du=(n-1)t^{n-2}\,\mathrm dt

\mathrm dv=e^{-t}\,\mathrm dt\implies v=-e^{-t}

Then

\displaystyle\Gamma(n)=-t^{n-1}e^{-t}\bigg|_0^\infty+(n-1)\int_0^\infty t^{n-2}e^{-t}\,\mathrm dt

We have

\displaystyle\lim_{t\to\infty}\frac{t^{n-1}}{e^t}=0

and so

\Gamma(n)=(n-1)\displaystyle\int_0^\infty t^{n-2}e^{-t}\,\mathrm dt=(n-1)\Gamma(n-1)

or, replacing n\to n+1, \Gamma(n+1)=n\Gamma(n).

b. From the above recursive relation, we find

\Gamma(n+1)=n\Gamma(n)=n(n-1)\Gamma(n-1)=n(n-1)(n-2)\Gamma(n-2)=\cdots=n(n-1)(n-2)\cdots2\cdot1\Gamma(1)

Now,

\Gamma(1)=\displaystyle\int_0^\infty e^{-t}\,\mathrm dt=1

and so we're left with \Gamma(n+1)=n!.

c. Using the previous result, we find \Gamma(5)=4!=24.

d. If the question is asking to find \Gamma(12), then you can just use the same approach as in (c).

But if you're supposed to find \Gamma\left(\frac12\right), we have

\displaystyle\Gamma\left(\frac12\right)=\int_0^\infty t^{-1/2}e^{-t}\,\mathrm dt

Substitute

u=t^{1/2}\implies u^2=t\implies 2u\,\mathrm du=\mathrm dt

Then

\displaystyle\Gamma\left(\frac12\right)=\int_0^\infty\frac1ue^{-u^2}(2u\,\mathrm du)=2\int_0^\infty e^{-u^2}\,\mathrm du=\frac{2\sqrt\pi}2=\sqrt\pi

You might be interested in
Don't guess!
Alja [10]

Answer:

The correct answer would be J; \frac{1}{2} r + 2;  

Step-by-step explanation:

Using the algebra;

\frac{1}{2}  * r = \frac{1}{2} r

\frac{1}{2} * 4 = 2

Altogether this would give "J" as the answer.

Hope this helps!

7 0
3 years ago
Read 2 more answers
Pls help me with this i have a 0 in math
Lorico [155]

Answer:

(3−5)(+1)

Step-by-step explanation:

I hope this helps you and please mark me as brainliest

6 0
3 years ago
Read 2 more answers
Dexter was asked to mix the punch for his sister’s birthday party. He has to figure out how much water and pure concentrate to a
Zigmanuir [339]

Answer:

  6 liters

Step-by-step explanation:

Let x represent the amount of pure concentrate used to make the punch. Then the total amount of concentrate in the punch is ...

 0.20·10 +1.00x = 0.50(10+x)

  x +2 = 0.5x +5 . . . . . eliminate parentheses

  0.5x = 3 . . . . . . . . . . .subtract 2+0.5x

  x = 6 . . . . . . . . . . . . . multiply by 2

6 liters of pure concentrate must be added to make a 50% punch.

5 0
3 years ago
Select all of the following true statements if R = real numbers, I = integers, and W = {0, 1, 2, ...}.
Vinvika [58]

Answer and Step-by-step explanation:

We will begin to solve this problem by defining first what the sets' elements really are.

R consists of real numbers. This means that this set contains all the numbers, rational or not.

Z is composed of whole numbers. Integers include all negative and positive numbers as well as zero (it's basically a set of whole numbers and their negated values).

W, on the other hand, has 0,1,2, and its elements are onward. Those numbers are referred to as whole numbers.

W ⊂ Z is TRUE. Z contains all the numbers as stated earlier, and W is a subset of it.

R ⊂ W is FALSE. Not all numbers are complete numbers. Complete numbers must be rational and represented fractionless. These requirements are not met by those real numbers.

0 ∈ Z is TRUE.  Zero is just an integer so it is a component of Z.

∅ ⊂ R is TRUE. A set i.e null be R subset, and each and every set is a general set. Moreover, there were not single elements in a null set, so it spontaneous became a non empty set subset through description as there is no element of R.

{0,1,2,...} ⊆ W is TRUE. The set on the left is precisely what is specified in the statement for problem for W. (The bar below the subset symbol simply implies that the subset is not rigid, because the set on the left may be equal to the set on the right. Without it, the argument would be incorrect, because a strict subset needs that the two sets not be identical).

-2 ∈ W is FALSE. W's only made up of whole numbers and not their negated equivalents.

4 0
3 years ago
What is the value of the underlined digit 28
joja [24]
The 2 is 20 and the 8 is 8 tens and ones your welcome
5 0
3 years ago
Other questions:
  • 23 hours after 8:00 pm
    11·2 answers
  • If there are 1200 calories in 8 ounces of hot fudge how many calories are in 3 ounces of hot fudge??
    13·1 answer
  • Substitution property of equality if y=-5 and 7x+y=11, then____. A. 7(-5) + y=11 B.7x-5=11 C.7x+5=11 D.-5+y=11
    8·1 answer
  • Equation practice with angle
    11·1 answer
  • What are 9's multiple
    12·2 answers
  • Brainliest solve 1-11 zoom in for 9 & 11
    10·1 answer
  • What is the answer I need help pls
    11·1 answer
  • Please I need help with this 3.1 (p-2.7)
    5·1 answer
  • I only have an hour left!
    14·1 answer
  • ANSWER QUICKLY! Brainliest if you get the correct answer
    9·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!