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2.12 The Error Function 203

EXERCISES

1. Show that erf(−z)=−erf(z),thatis,thaterfisanoddfunction.


2. Carry out the integration indicated in Eq. (8).


3. Verify these properties of the complementary error function:

a.

d

dzerfc(z)=−e

−z^2 √^2

π;

b.erfc( 0 )=1;

c. zlim→∞erfc(z)=0;

d.z→−∞lim erfc(z)=2;

e.erfc(z)is neither even nor odd.

4. Verify by differentiating thatu(x,t)=erf(x/

√

4 kt)satisfies the heat equa-

tion (1).

5. Ve r i f y t h a tu(x,t)=erf(x/

√

4 kt)satisfies the initial condition

u(x,t)=

{ 1 , 0 <x,

− 1 , x<0.

6. In probability and statistics, thenormal,orGaussian, probability density
   function is defined as

f(z)=

1

√

2 π

e−z^2 /^2 , −∞<z<∞,

and the cumulative distribution function is

(x)=

∫x

−∞

f(z)dz.

Show that the cumulative distribution function and the error function are

related by(x)=[1+erf(x/

√

2 )]/2.

7. Express this integral in terms of the error function:

I(x)=

∫ e−x

√xdx.

8. Use error functions to solve the problem
   ∂^2 u
   ∂x^2 =

1

k

∂u

∂t,^0 <x,^0 <t,