Miscellaneous Exercises 391
9.Find the transform of the solution of
∂^2 u
∂x^2=∂u
∂t, 0 <x, 0 <t,
u( 0 ,t)= 0 , 0 <t,
u(x, 0 )= 1 , 0 <x,
u(x,t)bounded asx→∞.10.At the end of Section 2.12, the problem in Exercise 9 was solved by other
means. Use this fact to identify
1
s
(
1 −e−√sx)
=L[
erf( x
√
4 t)]
and
1
se−√sx
=L[
1 −erf( x
√
4 t)]
.
(The latter function is called thecomplementary error function,defined
by erfc(q)≡ 1 −erf(q).)11.Find the function oftwhose Laplace transform is
F(s)=e−x√s
.12.Using the definition of sinh in terms of exponentials and a geometric
series, show that
sinh(√sx)
sinh(√s) =∑∞
n= 0(
e−√s( 2 n+ 1 −x)
−e−√s( 2 n+ 1 +x))
.13.Use the series in Exercise 12 to find a solution of the problem in Exer-
cise 6 in terms of complementary error functions.
14.Show the following relation by using Exercise 11 and differentiating with
respect tos.
L[
√^1
πtexp(
−k^2
4 t)]
=√^1
se−k√s
.15.Find the Laplace transform of the odd periodic extension of the function
f(t)=π−t, 0 <t<π,by transforming its Fourier series term by term.