Mathematical Tools for Physics

1—Basic Stuff 25

1.11 What is the integral

∫∞

−∞dtt

ne−t^2 ifn=− 1 orn=− 2? [Careful!, no conclusion-jumping allowed]

1.12 Sketch a graph of the error function. In particular, what is it’s behavior for smallxand for largex, both

positive and negative? Note: “small” doesn’t mean zero. First draw a sketch of the integrande−t

2
and from
that you can (graphically) estimateerf(x)for smallx. Compare this to the short table in Eq. ( 11 ).

1.13 Put a parameterαinto the defining integral for the error function, so it hase−αt

2

. Differentiate and show
that ∫x

0

dtt^2 e−t

2

=

√

π

4

erf(x)−

1

2

xe−x

2

As a check, does this agree with the previous result forx=∞, Eq. ( 10 )?

1.14 Use partial integration or other means to derive the identityxΓ(x) = Γ(x+ 1).

1.15 What is the Gamma function ofx=− 1 / 2 ,− 3 / 2 ,− 5 / 2? Explain why the original definition ofΓin terms
of the integral won’t work here. Demonstrate why Eq. ( 12 ) converges for allx > 0 but does not converge for
x≤ 0.

1.16 What is the Gamma function forxnear to 1? near 0? near− 1? − 2? − 3? Now sketch a graph of the
Gamma function from− 3 through positive values.

1.17 Show how to express the integral for arbitrary positivex

∫∞

0

dttxe−t

2

in terms of the Gamma function. Ispositivexthe best constraint here or can you do a touch better?

1.18 The derivative of the Gamma function atx= 1isΓ′(1) =− 0 .5772 =−γ. The numberγis called Euler’s
constant, and likeπoreit’s another number that simply shows up regularly. What isΓ′(2)? What isΓ′(3)?