I don't understand why the four transform F{(a/sqrt(pi))*e^-a^2*x^2)}=e^-alpha^2/4a^2. Can you please explain it to me . Thank you
I don't understand why the four transform F{(a/sqrt(pi))*e^-a^2*x^2)}=e^-alpha^2/4a^2. Can you please explain it to me . Thank you
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I don't understand why the four transform F{(a/sqrt(pi))*e^-a^2*x^2)}=e^-alpha^2/4a^2. Can you please explain it to me . Thank you
![### Claim
\[
\sqrt{ \frac{e^{-\frac{x^2}{4kt}}}{2\pi kt} } = e^{-k v^2 t}
\]
### Reason:
- **We Know**
\[
\sqrt{ \frac{a}{\pi} e^{-a^2 x^2} } = e^{-\frac{v^2}{4a^2}}
\]
- **Set**
\[
kt = \frac{1}{4a^2}
\]
Therefore,
\[
a^2 = \frac{1}{4kt}, \quad a = \frac{1}{2\sqrt{kt}}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca708747-373f-4dbe-b127-10492ca0e68f%2F781d7040-f7f1-4562-bf8f-c5be49059c38%2F3e5aim_processed.png&w=3840&q=75)
Transcribed Image Text:### Claim
\[
\sqrt{ \frac{e^{-\frac{x^2}{4kt}}}{2\pi kt} } = e^{-k v^2 t}
\]
### Reason:
- **We Know**
\[
\sqrt{ \frac{a}{\pi} e^{-a^2 x^2} } = e^{-\frac{v^2}{4a^2}}
\]
- **Set**
\[
kt = \frac{1}{4a^2}
\]
Therefore,
\[
a^2 = \frac{1}{4kt}, \quad a = \frac{1}{2\sqrt{kt}}
\]
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