Hi PF! I was wondering if you could clarify something for me. Specifically, I am solving the heat equation ##u_t = u_{xx}## subject to ##| u(\pm \infty , t ) | < \infty##. Now this implies a solution of sines and cosines times an exponential. Since we have a linear PDE, we may superimpose each solution (we have infinitely many since ##\lambda##, the separation constant, need only be positive or negative, depending on how we define it).(adsbygoogle = window.adsbygoogle || []).push({});

Since ##\lambda## is continuous we may sum via the Reimann integral, as $$\int_0^\infty c_1(\lambda ) \sin (\sqrt{\lambda} x) \exp (- \lambda k t) d \lambda$$. There would also be a cosine expression. My question is, where does the ##d \lambda## come from? Can someone please explain?

Thanks so much!

Josh

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# Fourier Transform

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